1 Introduction

The numerical solution of optimal control problems where the control function is of bang-bang or bang-singular type, often is a challenging problem. In the literature, one can find numerous examples of successful treatment for various applications, e.g. in [30, 31, 33, 47, 48]. More references are provided in [6, 14].

The approaches include direct discretizations as well as indirect methods like shooting approaches, partly in combination with so-called arc–parameterization methods; cf. [6, 28, 34] for examples. The latter usually require an initial hypothesis on the principal control structure, i.e., the number and kind of subsequent bang-bang or singular control arcs. It was this fact which has aroused re-newed interest in classical discretization methods like Euler or Runge–Kutta schemes and their use for detecting control structures.

For optimal control problems with continuous control regimes, convergence results are found e.g. in [12, 13, 32].  In case of bang-bang control behavior, first convergence results have been obtained for Runge–Kutta methods in [45, 46]. Recently, convergence of the Euler method has been obtained under appropriate bang-bang regularity and second-order optimality conditions in [1–4]. An extension to bang-bang switches of weaker regularity is addressed in [24].

Concerning the theoretical analysis of numerical methods in bang-singular control case, important advances have been achieved in deriving optimality conditions [5, 9, 20, 21, 37–39, 44], but also in stability investigation of solutions [14, 15, 40]. For the latter, the shooting method has been utilized e.g. in [47], and its convergence in combination with an arc parameterization was proven in [6]. An example for using Euler’s discretization can be found in [1] but the error analysis therein was mainly taylored for bang-bang type optimal controls and remained fragmentary for the bang-singular case. With the present paper, we make an attempt to show that a particular Euler discretization of the first-order system of necessary optimality conditions (cf. [15]) converges, and enables us to detect the control structure with an accuracy of order one of mesh size. The analysis is so far restricted to scalar-valued controls, fixed initial states, and to semilinear state systems (where the control input enters the system linearly with constant coefficients).

1.1 Plan of the paper

After the Introduction, the problem and main assumptions are stated in Sect. 2. The conditions include growth restrictions on problem data, a structural assumption, and certain second-order optimality condition. Section 3 describes the discretized problem in variational inequality form including a right-hand side (rhs) perturbation, and analyzes the existence and uniqueness of solutions. Using an appropriate topology in the discrete function space, Robinson’s concept of strongly regular generalized equations [41–43] is applied in Sect. 4. The main Lipschitz stability result (Theorem 1) is obtained by adapting the fixpoint iteration from [42] to the problem class. In Sect. 5, the properties of the controls are reconsidered in case when the rhs term vanishes. For sufficiently small stepsize parameters h, the discrete control is proved to have the same principal bang-singular-bang structure as the exact solution of the continuous problem. In \(L_{\infty }\times L_{1}\) topology, the discrete state-control pairs converge to the reference solution as  O(h)  (see Theorem 2). Finally, a numerical example illustrates the applicability of the approach.

1.2 Notation

Let be given \(N\gg 1\). With \(h=1/N\), find the set of equidistant nodes \(t_{i}=\,ih,\;i=0,\ldots ,N\),  on [0, 1]. A (multi-)vector \(v=(v_{0},\ldots ,v_{N-1})\in \mathbb {R}^{kN}, \;v_{i}\in R^{k},\) can be given the interpretation of a piecewise continuous function with constant values \(v_{i}\) on the interval \(\omega _{i}=(t_{i},t_{i+1})\) whereas \(y=(y_{0},\ldots ,y_{N})\in \mathbb {R}^{k(N+1)}\) corresponds, e.g., to a piecewise linear function with nodes \(y(t_{i})=y_{i}\). Thus, the related function spaces denoted by \(V^{h}\subset \,L_{\infty }, \,Y^{h}\subset \,W_{\infty }^{1}\) are finite-dimensional and their elements will be identified with the coefficient vectors \(v\in \mathbb {R}^{kN},\,y\in \mathbb {R}^{k(N+1)}\).   In \(Y^{h}\subset \, L_{\infty }\subset \, L_{2}\), introduce the following norms

$$\begin{aligned} \Vert y\Vert _{\infty }=\,\max _{0\le i\le N}|y_{i}|,\qquad \Vert y\Vert _{(2)}=\,\left( |y_{N}|^{2}+\,h\, \sum _{i=0}^{N-1}|y_{i}|^{2}\right) ^{1/2}. \end{aligned}$$
(1)

Analogously, set \(\Vert v\Vert _{\infty }=\,\max _{0\le i\le N-1}|v_{i}|, \Vert v\Vert _{p}=\,\left( h\,\sum _{i=0}^{N-1}|v_{i}|^{p}\right) ^{1/p}\)  for \(v\in V^{h}\subset L_{\infty }\subset L_{p}, 1\le p <\infty \). In the above definitions, \(|\cdot |\) stands for the standard Euclidean vector norm in \(\mathbb {R}^{k}\).  If functions are restricted to a certain interval \(I\subset [0,1]\), the modified norms are written as  \(\Vert \cdot \Vert _{(p,I)}\).

The scalar product in \(V^{h}\) will be defined as the related \(L_{2}\) scalar product by  \((v,v')\,:=\,h\sum _{i=0}^{N-1}v_{i}^{T}v'_{i}\)  with the matrix notation \(a^{T}b\) for the standard scalar product of two column vectors \(a,b\in \mathbb {R}^{k}\).   Finally, for \(y\in \mathbb {R}^{N+1}\) define the finite difference operators \(\Delta ^{1}y\in \mathbb {R}^{N}\) and \(\Delta ^{2}y\in \mathbb {R}^{N-1}\),

$$\begin{aligned} (\Delta ^{1}y)_{i}=\,h^{-1}(y_{i+1}-y_{i}),\quad (\Delta ^{2}y)_{i}=\,h^{-2}(y_{i+1}-2y_{i}+y_{i-1}),\;\;i\le N-1. \end{aligned}$$

2 Statement of the problem: assumptions

Consider semilinear control problems with scalar-valued control input and prescribed initial state,

figure a

subject to

$$\begin{aligned} \qquad \quad&\dot{x}(t)\;=f(x(t))\,+\,B\,u(t) \quad \text { a.e. in }\;[0,1], \qquad \qquad \end{aligned}$$
(3)
$$\begin{aligned}&x(0)\,=\,a,\;\;\end{aligned}$$
(4)
$$\begin{aligned}&\underline{\alpha }\;\le \,u(t)\,\le \; \overline{\alpha }, \,\quad \text {a.e. in}\;[0,1]\,, \end{aligned}$$
(5)
$$\begin{aligned}&x\in W_\infty ^1(0,1;\mathbb {R}^n),\;\; u\in L_\infty (0,1;\mathbb {R}). \end{aligned}$$
(6)

For sake of simplicity, we assume  \(\underline{\alpha }=-1,\;\overline{\alpha }=\,1\). Notice that, without changing the problem type, general control bounds transform easily to this case, and all results will be invariant w.r.t. possible control rescaling.

Assume that the data functions satisfy

(H0)   (smoothness assumption)

The functions \(f,\,k\) depend smoothly on \(x\in \mathbb {R}^{n}\):  \(f,k\in C^{3,1}\),  and there exist constants \(c,\,L\) independent of x such that

$$\begin{aligned} |f(x)|\;\le \; c\,+\,L\,|x| \qquad \text{ for } \text{ all }\;\; x\in \mathbb {R}^{n}. \end{aligned}$$

Moreover, \(B\in \mathbb {R}^n\) is constant.

Define the Pontryagin function \(H:\,\mathbb {R}^{n}\times \mathbb {R}\times \mathbb {R}^{n}\rightarrow \,\mathbb {R}\) by

$$\begin{aligned} H(x,u,p)\,=\,p^Tf(x)\,+\,p^TB\,u. \end{aligned}$$

Then, for the given normal case, Pontryagin’s Maximum Principle can be expressed in form of a variational inequality (or: inclusion)

figure b

for almost every \(t\in [0,1]\). The set \(N_+\) stands for the normal cone to \(\mathbb {R}_{+}\). Conditions (9), (10) can be equivalently written in terms of the switching function \(\sigma (\cdot )\,=\,B^{T}p(\cdot )\) and its positive resp. negative parts \([\sigma ]_{\pm }\): for \(\mu _{1}=[\sigma ]_{+},\, \mu _{2}=[\sigma ]_{-}\) and \(\sigma =[\sigma ]_{+}-\,[\sigma ]_{-}\), the relations yield the complementarity system

$$\begin{aligned}{}[\sigma ]_{+},\,[\sigma ]_{-}\ge \,0,\quad -1\le u\le \,1,\quad (u-1)\,[\sigma ]_{+} =\,(u+1)\,[\sigma ]_{-}=\,0. \end{aligned}$$

Assume that \((x^{0},u^{0})\in W_{\infty }^{1}\times L_{\infty }\) is a (strong) local minimizer of (CP), i.e., there exists an \(\epsilon >0\) such that

$$\begin{aligned} J(x,u)\,-\,J(x^{0},u^{0})\;\ge \;0 \end{aligned}$$

for all admissible pairs (x, u) satisfying \(\Vert x-x^{0}\Vert _{\infty }<\,\epsilon \).

Then, the adjoint and multiplier functions \(p^{0},\mu _{1}^{0},\mu _{2}^{0}\) associated to \((x^{0},u^{0})\) are uniquely determined by (VI) and belong to \(W_{\infty }^{1}\). Further,  \(\sigma ^{0}=B^{T}p^{0}\in \;C^{1}\), and \(\dot{\sigma }^{0}=-B^{T}H_{x}^{0}=-B^{T}\nabla f(x^{0})^{T}p^{0} \;\in \,W_{\infty }^{1}\).   For  \(\ddot{\sigma }^{0}\)  find

$$\begin{aligned} \ddot{\sigma }^{0}=\,P+{u^{0}}R,\quad P=B^{T}(\nabla f(x^{0})^{T}H^{0}_{x}-H^{0}_{xx}f^{0}),\;\; R=-B^{T}H^{0}_{xx}B, \end{aligned}$$
(11)

with functions \(P,\,R\) belonging to \(W^{1}_{\infty }\).

Optimality conditions for (CP) in case of a bang-singular-bang control switching structure have been derived in [5], see also [7–10, 26, 36, 39, 44]. Allover the paper, we use the following assumptions:

(H1)   (strict structural assumption)

The function \(u^{0}\) is of strict bang-singular-bang structure, i.e., there exist points \(\tau _{s},\tau _{s}'\) with \(0<\tau _{s}<\tau _{s}'<1\) and a positive constant m such that

$$\begin{aligned} u^{0}(t)\,=\,\left\{ \begin{array}{cl@{\quad }l} u^{1}&{} \text{ if }&{}\quad 0\le t<\tau _{s},\\ u^{0}_{s}(t)\quad &{}\text{ if }&{}\quad \tau _{s}<t<\tau _{s}',\\ u^{2}&{} \text{ if }&{}\quad \tau _{s}'<t\le 1, \end{array} \right. \end{aligned}$$

with constants \(u^{1},u^{2}\in \{-1,1\}\), and \(|u^{0}_{s}(t)|\le \,1-m\) for a.a. t on the singular arc \( [\tau _{s},\tau _{s}']\).   Moreover, \(\sigma ^{0}\not = 0\) on bang arcs \([0,\tau _{s})\cup (\tau _{s}',1]\).

For given \(\beta >0\), define the sets \(J_{\beta }=\{\,t\in [0,1]:\;|\sigma ^{0}(t)|\,\ge \,\beta >0\,\}\) and \(I_{\beta }=(0,1)\backslash J_{\beta }\).  Then the singular arc \([\tau _{s},\tau _{s}']\) is equal to  \(I_{0}=\bigcap _{\beta >0}I_{\beta }\).

(H2)   (strong second-order optimality condition)

Let \(\varOmega =\varOmega (v,y,b)\) denote the quadratic form

$$\begin{aligned} \varOmega (v,y,b)\,=\,z(1)^{T}K\,z(1)\,+\,\int _{0}^{1}z(t)^{T}Q(t)z(t)\,dt \end{aligned}$$
(12)

with  \(K=\nabla ^{2}k(x^{0}(1)), \;Q=\nabla ^{2}_{xx}H(x^{0},u^{0},p^{0})\), and \(v\in L_{2}(0,1;\mathbb {R}), y\in W_{2}^{1}(0,1;\mathbb {R}),\, z\in W_{2}^{1}(0,1;\mathbb {R}^{n})\) satisfying

$$\begin{aligned} y(t)= & {} \int _{0}^{t}v(s)\,ds,\qquad \quad y(1)=:\,b,\\ \dot{z}(t)= & {} A(t)z(t)\,+\,Bv(t), z(0)\,=\,0,\qquad A=\nabla f(x^{0}).\nonumber \end{aligned}$$
(13)

There exist constants \(\beta >0,\,m>0\) such that

$$\begin{aligned} \varOmega (v,y,b)\,\ge \,m\,\left( \Vert y\Vert ^{2}_{2}+b^{2}\right) \end{aligned}$$

for all \(v\in L_{2}\) with \(v\,=\,0\) on \(J_{\beta }\), and \((y,b)\in W_{2}^{1}\times \mathbb {R}\) given by (13).

(The formulation follows [15] but has been adapted to the special case of a semilinear state equation where both \(\nabla ^{2}_{uu}H\) and \(\nabla ^{2}_{xu}H\) vanish.) Without loss of generality, one can use the same constant m in both (H1), (H2) when shrinking it if necessary.

In [5, 9] it was proved that, under conditions (H1), (H2), the extremal \((x^{0},u^{0})\) is a strict Pontryagin minimum [35] for (CP). Moreover, the following relations hold:

$$\begin{aligned}&R\,\ge \, m\,>\,0\quad \qquad \text{ on }\quad I_{\beta }, \end{aligned}$$
(14)
$$\begin{aligned}&|\ddot{\sigma }(\tau _{s}-0)|\,\ge \,m^{2}>\;0,\;\; |\ddot{\sigma }(\tau '_{s}+0)|\,\ge \,m^{2}>\;0. \end{aligned}$$
(15)

Indeed, (14) had been proved in [14, Sect. 3.1]; cf. also [37]. While \(\ddot{\sigma }(\tau _{s}+0)=0\), conclude \(|\ddot{\sigma }(\tau _{s}-0)|=|R\cdot (u^0(\tau _s+0)-u^0(\tau _s+0))| \ge \,m^2\) from (11), and from jump estimates on \(u^0\) implied by (H1). Hence (15) is obtained, too.

It follows that the control \(u^{0}\) has a singular arc of order one (cf. [29, 49]) on \([\tau _{s},\tau '_{s}]\) and coincides there with  \(u^{0}_{s}=-P/R\).  If \(\beta \) is taken sufficiently small then, in addition, \(I_{\beta }\) is an open interval.

3 Variational inequality discretization

Let the interval [0, 1] be divided into subintervals \(\omega _{i}=[t_{i},t_{i+1}], \,0\le i< N\), of equal length \(h=1/N, \;N\gg 1\). Consider the following discretization of (VI):

figure c

for \(i=0,\ldots ,N-1\). The formulation includes a right-hand side (or: rhs) perturbation \(\delta =(\delta _{1},\delta _{2},\delta _{3})\) of dimension \(l=n(2N+1)+N\).

For \(\xi =(p,x,u,\mu )\in C=\mathbb {R}^{n(N+1)}\times \mathbb {R}^{n(N+1)}\times \mathbb {R}^{N}\times \mathbb {R}^{2N}_{+}\) and given \({\delta }\in \mathbb {R}^{l}\),  (VI\(^{h}_{\delta }\)) can be written in abstract form as

$$\begin{aligned} \bar{\delta }\;\in \;\psi (\xi )\,+\,N_{C}(\xi ), \end{aligned}$$
(20)

where \(N_{C}(\xi )\) denotes the normal cone to the set C at point \(\xi \), and \(\delta \) is completed to \(\bar{\delta }\) by zero entries \(\delta _{1N}\in R^{n}\) and \(\delta _{4}\in R^{2N}\). The dimension of the vectors \(\xi \) and \(\bar{\delta }\) is \(d=2n(N+1)+3N\).  For each \(h=1/N,\;N>1\), it follows from (H0) that \(\psi :\,\mathbb {R}^{d}\rightarrow \,\mathbb {R}^{d}\) is at least twice continuously differentiable.

Lemma 1

For each \(N>1, \,h=1/N\), and arbitrary \(\delta \in R^{l}\),  (\(\hbox {VI}_{\delta }^{h}\)) has a solution  \(\xi \in C\). There exists a constant \(\bar{r}>0\) independent of h and \(\delta \), such that \(\xi \) is bounded by   \(\Vert \xi \Vert _{\infty }\le \,\bar{r}(1+\Vert \delta \Vert _{\infty })\). Moreover,   \(\Vert u\Vert _{\infty }+\Vert x\Vert _{\infty }+\Vert p\Vert _{\infty } \le \,\bar{r}(1+\Vert \delta _{1}\Vert _{2}+\Vert \delta _{2}\Vert _{(2)})\) in the norms given by (1).

Proof

Any prospective solution of the variational inequality (20) satisfies \(\Vert u\Vert _{\infty }\le \,1\). Under Assumption (H0), for \(i \ge 0\)

$$\begin{aligned} |x_{i+1}|\,\le \, |x_{i}|\,+\,h\,\left| f(x_{i})+Bu_{i}+\delta _{1i}\right| \,\le \,(1+Lh) |x_{i}|\,+\,h\,(c+|B|+|\delta _{1i}|) \end{aligned}$$

so that, in analogy to Gronwall’s Lemma [23] (or Lemma 14, Appendix), the estimate

$$\begin{aligned} |x_{i}|\,\le \,e^{L}\left( |a|\,+\,c+|B|\right) \,+\,e^{L}\Vert \delta _{1}\Vert _{2} \,\le \,\bar{r}_{x}(1+\Vert \delta _{1}\Vert _{2}) \end{aligned}$$

follows. All constants herein are independent of h. Further, from (17) we backwards obtain similar bounds for p. In order to estimate \(\mu \), it should be noticed that (19) yields \(\mu _{1},\mu _{2}\ge \,0\) and \(\mu _{1i}\cdot \,\mu _{2i}=\,0\) for all i. Consequently, \(\mu _{1i}=\left[ B^{T}p_{i}-\delta _{3i}\right] _{+},\, \mu _{2i}=\left[ B^{T}p_{i}-\delta _{3i}\right] _{-}\) are bounded, e.g., by \(\bar{r}_{\mu }(1+\Vert \delta \Vert _{\infty })\) with constant \(r_{\mu }\) independent of h, and the desired estimate for \(\xi \) follows.

Using the above information, for given \(h,\delta \) one can restrict (VI\(^{h}_{\delta }\)) or (20) to \(\xi \in C_{\delta }= C \cap \{\xi :\;\Vert \xi \Vert _{\infty }\le \bar{r}(1+\Vert \delta \Vert _{\infty })\} \subset \mathbb {R}^{d}\). As far as \(C_{\delta }\) is non-empty, convex and compact, and \(\psi \) as an operator from \(C_{\delta }\) to \(\mathbb {R}^{d}\) is continuous, one can apply the existence result from Theorem 3.1 [27, Chap. 1] for variational inequalities in finite dimensions: for compact sets, the assertion follows directly by Brouwer’s Fixed-Point Theorem. \(\square \)

Using the solution \((x^{0},u^{0},p^{0},\mu ^{0})\) of the continuous problem (VI), one can find a reference solution for (\(\hbox {VI}_{\delta }^{h}\)) with appropriately chosen rhs term \(\delta =\tilde{\delta }^{h}\). To this aim, consider piecewise linear interpolations \(\tilde{x}^{h}, \tilde{p}^{h}\) with node values \(\tilde{x}^{h}(t_{i})=\tilde{x}^{h}_{i}=\,x^{0}(t_{i}),\)   \(\tilde{p}^{h}(t_{i})=p^{0}(t_{i})\) and constant \(\dot{\tilde{x}}^{h}, \,\dot{\tilde{p}}^{h}\) on each interval \(\omega _{i},\,i=0,\ldots ,N-1\). Further, define \(\tilde{u}^{h}\) as as a piecewise continuous function with constant values \(\tilde{u}^{h}_{i}\) on \(\omega _{i},\;i=0,\ldots ,N-1,\) by

$$\begin{aligned} \left. \tilde{u}^{h}\right| _{\omega _{i}}\equiv \,\tilde{u}^{h}_{i}=\, h^{-1}\int _{\omega _{i}}u^{0}(t)\,dt. \end{aligned}$$
(21)

If we denote by \(k,k'\) indices such that \(t_{k}\le \tau _{s}<t_{k+1},\,t_{k'}<\tau _{s}'\le t_{k'+1}\) then \(\tilde{u}^{h}_{i}\) takes constant extremal values \(u^{1}\) (or \({u}^{2}\) resp.) for \(i\le k-1\) (or \(i\ge k'+1\)), but has singular values \(\tilde{u}^{h}_{i}\in (0,1)\) for \(k\le i\le k'\).

For \(S_{\tau }:=\{k,k+1,k',k'+1\}\), introduce

$$\begin{aligned} \Vert \cdot \Vert _{(\infty ,\tau )}=\;\max \{|\cdot |_{i}:\; i\not \in S_{\tau }\}. \end{aligned}$$
(22)

Lemma 2

Let \(h=1/N\) be sufficiently small. Then the functions \(\tilde{x}^{h},\tilde{u}^{h}\) and \(\tilde{p}^{h}\) together with \(\tilde{\mu }^{h}\) given by

$$\begin{aligned} \tilde{\mu }^{h}_{1i}=\,\left[ \tilde{\sigma }^{h}_{i}\right] _{+},\quad \tilde{\mu }^{h}_{2i}\;=\;\left[ \tilde{\sigma }^{h}_{i}\right] _{-} \end{aligned}$$

with the definition

$$\begin{aligned} \tilde{\sigma }^{h}_{i}=\,\sigma ^{\tau }(t_{i}), \qquad \sigma ^{\tau }(t)\,=\, \left\{ \begin{array}{lll}\sigma ^{0}(t+\tau _{s}-t_{k})&{}\quad \text{ if }&{}\quad t\le t_{k},\\ \;0&{}\quad \text{ if }&{}\quad t_{k}<t\le t_{k'},\\ \sigma ^{0}(t+\tau '_{s}-t_{k'})&{}\quad \text{ if } &{}\quad t>t_{k'}, \end{array}\right. \end{aligned}$$

solve (\(\hbox {VI}_{\delta }^{h})\) for some \(\delta =\tilde{\delta }^{h}\) satisfying \(|\tilde{\delta }_{1i}^{h}|+\,|\tilde{\delta }_{2i}^{h}| +\,|\tilde{\delta }_{3i}^{h}|=\) O(h), together with \(\tilde{\delta }^{h}_{2N}=0\).   Moreover, \(|(\Delta ^{1}\tilde{\delta }_{3}^{h})_{i}|\, =\mathrm{O}(h)\) uniformly for \(\,i<N, \Vert \Delta ^{2}\tilde{\delta }_{3}^{h}\Vert _{2}= \mathrm{O}(\sqrt{h})\) and  \(\Vert \Delta ^{1}\tilde{\delta }^{h}_{2}\Vert _{(\infty ,\tau )}+ \Vert \Delta ^{2}\tilde{\delta }^{h}_{3}\Vert _{(\infty ,\tau )}=\mathrm{O}(h)\).

The proof is given in Appendix.

Lemma 3

Under Assumptions (H1) and (H2), the discrete switching function \(\tilde{\sigma }^{h}\) satisfies \(\Vert \Delta ^{1}\tilde{\sigma }^{h}\Vert _{2}+\Vert \Delta ^{2}\tilde{\sigma }^{h}\Vert _{2}\le c\) with a constant c independent of h. Further,

$$\begin{aligned} (\Delta ^{2}\tilde{\sigma }^{h})_{i}=\; \tilde{P}^{h}_{i}+\,R_{i}\tilde{u}^{h}_{i} \end{aligned}$$

for some \(\tilde{P}^{h}\) with the property \(\Vert \tilde{P}^{h}-P\Vert _{(\infty ,\tau )}=\mathrm{O}(h)\), and \((P_{i},R_{i})=(P(t_{i}),R(t_{i}))\).

Proof

By direct calculation, for \(\Delta ^{2}\tilde{\sigma }^{h}\) obtain

$$\begin{aligned} (\Delta ^{2}\tilde{\sigma }^{h})_{i}=\,\left\{ \begin{array}{lcl} (\Delta ^{2}\sigma ^{\tau })_{i}&{}\quad \text{ if }\, &{}\quad i\le k-1\;\;\text{ or }\;\;i\ge k'+1,\\ \ddot{\sigma }^{0}(t_{i})&{}\quad \text{ if }&{}\quad k+1\le i\le k'-1,\\ h^{-2}\sigma ^{0}(\tau _{s}-h)&{}\quad \text{ if }&{} \quad i=k,\\ h^{-2}\sigma ^{0}(\tau '_{s}+h)&{}\quad \text{ if }&{} \quad i=k'. \end{array}\right. \end{aligned}$$

Therefore, for all \(i\not \in \{k,k'\}\), we can write \((\Delta ^{2}\sigma ^{\tau })_{i}=\,\ddot{\sigma }^{0}(\theta _{i})\) for some \(\theta _{i}\) satisfying \(|\theta _{i}-\,t_{i}|=\hbox {O}(h)\) so that

$$\begin{aligned} (\Delta ^{2}\sigma ^{\tau })_{i}=\,P(\theta _{i})\,+\,R(\theta _{i})u^{0}(\theta _{i}). \end{aligned}$$

A closer look on the possible localization of \(\theta _{i}\) yields

$$\begin{aligned} u^{0}(\theta _{i})=\tilde{u}^{h}_{i}=u^{1}\;\;\text{ for }\;\;i\le k-1,\qquad u^{0}(\theta _{i})=\tilde{u}^{h}_{i}=u^{2}\;\;\text{ for }\;\;i\ge k'+1, \end{aligned}$$

and   \(u^{0}(\theta _{i})=u^{0}_{s}(t_{i})\,=\,-P_{i}/R_{i},\; k+1\le i\le k'-1\). For the latter, it follows from \(x^{0},p^{0},P,R\in W_{\infty }^{1}\) and condition (14) that \(|u^{0}_{s}(t_{i})-\tilde{u}^{h}_{i}|=\hbox {O}(h)\). Using further  \(|R(\theta _{i})-R_{i}|=\hbox {O}(h)\) and the boundedness of R, the desired representation and estimates are obtained. \(\square \)

4 Solution uniqueness and estimates for (VI\(^{h}_{\delta }\))

4.1 Linearized VI problem

Let the functions \(\tilde{x}^{h}, \tilde{p}^{h}, \tilde{u}^{h}\) and \(\tilde{\sigma }^{h}\) be given as in Sect. 3, (21) ff. For \(\xi =(p,x,u,\nu )\), the following linearized variational inequality related to (\(\hbox {VI}_{\delta }^{h}\)) will be considered:

$$\begin{aligned} \bar{\delta } \,\in \,T(\xi )\,+\,N_{C}(\xi ) \end{aligned}$$
(23)

where \(T: \,\mathbb {R}^{d}\rightarrow \,\mathbb {R}^{d}\) is the linearization of \((\psi -\bar{\delta })\) at \((\tilde{\xi }^{h},\tilde{\delta }^h)\),

$$\begin{aligned} T(\xi )=\psi (\tilde{\xi }^{h}) +\psi '(\tilde{\xi }^{h})(\xi -\tilde{\xi }^{h})-\bar{\delta }^h, \end{aligned}$$
(24)

and \(\bar{\delta }^h\in \mathbb {R}^d\) is the extension by zeros related to \(\tilde{\delta }^h\).

Denoting \(z_{i}=x_{i}-\,\tilde{x}^{h}_{i},\, q_{i}=\,p_{i}-\,\tilde{p}^{h}_{i}\) and \(v_{i}=\,u_{i}-\,\tilde{u}^{h}_{i}\), in detail the relations read as follows

figure d

for \(i=0,\ldots ,N-1\).  The coefficient matrices are

$$\begin{aligned} A_{i}=\,\nabla f(x^{0}(t_{i})), \quad \;\; Q_{i}=\,\nabla ^{2}_{xx}H^{0}[t_{i}],\qquad K\,=\,\nabla ^{2}k(x^{0}(1)). \end{aligned}$$

Notice that, for \(\delta =0\), the system (\(\hbox {LVI}^{h}_{\delta }\)) has the solution \((q,z,v,\nu )=(0,0,0,\tilde{\mu }^{h})\). In general, every solution (z, q) of (25), (26) can be represented as

$$\begin{aligned} z=S_{1}v+z^{h}_{\delta },\quad q=S_{2}v+q^{h}_{\delta }, \end{aligned}$$

where \((S_{1}v, S_{2}v)\) are the solutions for \(\delta =0\) written in terms of linear solution maps \(S_{1,2}\), but \((z^{h}_{\delta },q^{h}_{\delta })\) are related particular solutions for \(v=0,\) and given \(\delta _{1}\in \mathbb {R}^{Nn}, \delta _{2}\in \mathbb {R}^{(N+1)n}, \delta _{3}\in \mathbb {R}^{N}\) and \(q_{N}=\delta _{2N}-Kz^{h}_{\delta ,N}\). Inserting the expressions into (27) leads to

$$\begin{aligned} B^{T}q_{i}-\,\delta _{3i}=:\, -(C^{h}v)_{i}+\,r^{h}_{i}(\delta ) \end{aligned}$$
(29)

with \(r^{h}_{i}(\delta )=\,B^{T}q^{h}_{\delta ,i}-\,\delta _{3i}\) and \((C^{h}v)_{i}=-B^{T}(S_{2}v)_{i}\).

Lemma 4

Every solution \((q,z,v,\nu )\) of \((\hbox {LVI}_{\delta }^{h}\)) solves

figure e

for \(i=0,\ldots ,N-1\)  on \(W^{h}=\,\{v'\in \mathbb {R}^{N}:\;-1\le v'_{i}+\tilde{u}^{h}_{i}\le 1,\;i=0,\ldots ,N-1\}\). If, in addition, \(|\nu _{1i}-\tilde{\mu }^{h}_{1i}|+|\nu _{2i}- \tilde{\mu }^{h}_{2i}|<\beta \,/2\) on \(J_{\beta }\), and h is sufficiently small, then v solves (LVI’) on   \(W^{h}_{\beta }=\,\{v'\in W^{h}:\;\; v_{i}'=0\;\,\text{ for }\;\,t_{i}\in J_{\beta }\}\) where \(J_\beta =\{t\in [0,1]:\,\sigma ^0(t)\ge \beta \}\), and \(\beta \) is the constant from (H2).

The proof is a direct consequence of the complementarity relations (28).

4.2 Discrete Goh transformation

Define the vector \((y,\,\zeta ) \,\in \,\mathbb {R}^{N+1}\times \mathbb {R}^{n(N+1)}\) by

$$\begin{aligned} (\Delta ^{1}y)_{i}&=\,v_{i}\,=\,h^{-1}\int _{\omega _{i}}v(t)\,dt,\quad y_{0}=\,0, \end{aligned}$$
(30)
$$\begin{aligned} \zeta _{i}&= z_{i}-\,By_{i}. \end{aligned}$$
(31)

Inserting the expressions into (25), (26) yields

$$\begin{aligned} (\Delta ^{1}\zeta )_{i}= & {} (\Delta ^{1}z)_{i}-\,B(\Delta ^{1}y)_{i}\nonumber \\= & {} A_{i}z_{i} +\,\delta _{1i}\nonumber \\= & {} A_{i}\zeta _{i}+B^{1}_{i}y_{i} +\delta _{1i} \qquad \text{ with } \;B^{1}_{i}=A_{i}B; \end{aligned}$$
(32)
$$\begin{aligned} (\Delta ^{1}q)_{i}= & {} -A_{i+1}^{T}q_{i+1}-Q_{i+1}z_{i+1}+\,\delta _{2i} \nonumber \\= & {} -A_{i+1}^{T}q_{i+1}-Q_{i+1}\zeta _{i+1} -M_{i+1}^{T}y_{i+1}+\delta _{2i} \nonumber \\&\;\;\text{ with }\quad M_{i}^{T}=Q_{i}B. \end{aligned}$$
(33)

The boundary conditions transform into

$$\begin{aligned} \zeta _{0}=\,0,\quad y_{N}=:\,b,\;\;q_{N}=-K\,(\zeta _{N}+B\,b)+\delta _{2N}. \end{aligned}$$
(34)

The solutions of the transformed linearized state-adjoint system (32), (33) for \(\delta =0\) can be expressed by \(\zeta =Sy,\;q=\tilde{S}y+\tilde{W}b\) where \(S,\, \tilde{S}\) are linear solution operators to (32), (33). Thus, (27) turns into

$$\begin{aligned} -(\hat{C}^{h}y)_{i}+\,\tilde{\sigma }^{h}_{i}-\,\nu _{1i} +\,\nu _{2i}+\,r_{i}^{h}(\delta )=\,0 \end{aligned}$$
(35)

with the abbreviation  \((\hat{C}^{h}y)_{i}=-B^{T}(\tilde{S}y+\tilde{W}b)_{i}\), and the function \(r^{h}\) given by (29).

Lemma 5

Let the Assumptions (H0)–(H2) hold. If  \(h<\bar{h}\) with \(\bar{h}\) sufficiently small then \(\varOmega ^{h}(y)\,:=\,\left( C^{h}v,\,v\right) \) satisfies

$$\begin{aligned} \varOmega ^{h}(y)\,\ge \,\frac{m}{2}\,\left( \Vert y\Vert ^{2}_{2}+\,|y_{N}|^{2}\right) \end{aligned}$$
(36)

for all \(v\in \mathbb {R}^{N},\,y\in \mathbb {R}^{N+1}\) such that (30) holds together with \(v_{i}=0\) for \(t_{i}\in J_{\beta }\).

(As far as the constant m is taken from (H2) it is independent of N resp. h.)

Proof

Denote by \(\varPsi , \,\varPhi \) the discrete fundamental solutions of (32), (33),

$$\begin{aligned} \varPsi _{i+1}-\varPsi _{i}=\,hA_{i}\varPsi _{i},\quad \varPhi _{i+1}-\varPhi _{i}=\,-hA_{i+1}^{T}\varPhi _{i+1}, \;\;\varPsi _{0}=\,\varPhi _{-1}=\,I. \end{aligned}$$

Notice that,  for all \(i, \varPhi _{i-1}^{T}\varPsi _{i}=\varPsi _{i}^{T}\varPhi _{i-1}=\,I\). Then, for the discrete data \(\zeta ,\,q\) explicit formulas can be given:

$$\begin{aligned} \zeta _{i}= & {} (Sy)_{i} =\,h\,\varPsi _{i}\sum _{k=0}^{i-1}\varPhi _{k}^{T}B_{k}^{1}y_{k}, \\ q_{i}= & {} (\tilde{S}y)_{i}+\,\tilde{W}_{i}b\\= & {} -\varPhi _{i}\varPsi _{N+1}^{T}K(\zeta _{N}+Bb)\,+\, h\,\varPhi _{i}\sum _{k=i}^{N-1}\varPsi _{k+1}^{T}(Q_{k+1} \zeta _{k+1}+\,M_{k+1}^{T}y_{k+1}). \end{aligned}$$

Consequently, for some positive c independent of h,

$$\begin{aligned} \Vert \zeta \Vert _{\infty }+\,\Vert q\Vert _{\infty }\le \;c\,\left( \Vert y\Vert _{2}+|b|\right) . \end{aligned}$$

In order to prove the assertion of the lemma, start with \((\hat{C}^{h}y)_{i} =\,(C^{h}v)_{i} =-B^{T}q_{i}\); cf. (35). Denoting \(\tilde{y}_{i}=h\sum _{k=0}^{i-1}\tilde{v}_{k},\,i=0,\dots ,N\), the quadratic form transforms as follows:

$$\begin{aligned} \left( C^{h}v,\tilde{v}\right)= & {} -h\sum _{i=0}^{N-1}B^{T}q_{i}(\Delta ^{1}\tilde{y})_{i}= -\tilde{b}B^{T}q_{N}+\,h\sum _{i=1}^{N} \tilde{y}_{i}B^{T}(\Delta ^{1}q)_{i-1}\\= & {} \tilde{b}B^{T}K(\zeta _{N}+Bb)\,-\, h\sum _{i=1}^{N}\tilde{y}_{i}B^{T}(A_{i}^{T}q_{i}+ \,Q_{i}\zeta _{i}+M_{i}^{T}y_{i})\\= & {} \tilde{b}B^{T}K(\zeta _{N}+Bb)\,+\, h\sum _{i=1}^{N}\tilde{y}_{i}(R_{i}y_{i}-M_{i}\zeta _{i}\,-\, (B_{i}^{1})^{T}q_{i}). \end{aligned}$$

In the last sum on the right, one can substitute the explicit representation for \(q_{i}\) and obtain

$$\begin{aligned} - h\sum _{i=1}^{N}\tilde{y}_{i}(B_{i}^{1})^{T}q_{i}= & {} h\sum _{i=1}^{N}\tilde{y}_{i}(B_{i}^{1})^{T} \varPhi _{i}\varPsi _{N+1}^{T}K(\zeta _{N}+Bb)\\&\quad -\,h^{2}\sum _{i=0}^{N-1}\sum _{k=i}^{N-1} \tilde{y}_{i}(B_{i}^{1})^{T}\varPhi _{i}\varPsi _{k+1}^{T} (Q_{k+1}\zeta _{k+1}+M_{k+1}^{T}y_{k+1})\\= & {} \tilde{\zeta }_{N+1}^{T}K(\zeta _{N}+Bb) -h\sum _{k=0}^{N-1}\tilde{\zeta }_{k+1}^{T}(Q_{k+1}\zeta _{k+1}+M_{k+1}^{T}y_{k+1})\\= & {} \tilde{\zeta }_{N}^{T}K(\zeta _{N}+Bb)\,+\,\pi _{N}(y,\tilde{y}) \,-\,h\sum _{i=1}^{N}\tilde{\zeta }_{i}^{T}(Q_{i}\zeta _{i}+M_{i}^{T}y_{i}). \end{aligned}$$

For \(\tilde{y}=y\), the error term \(\pi _{N}=({\zeta }_{N+1}-{\zeta }_{N})^{T}K(\zeta _{N}+Bb)\) is bounded by

$$\begin{aligned} |\, \pi _{N}|\,\le \,h\,c_{\varOmega }\left( \Vert y\Vert _{2}^{2}+\, |b|^{2}\right) \end{aligned}$$

with some positive constant \(c=c_{\varOmega }\) independent of h.

The sum over the remaining expressions corresponds to a standard integral sum for the discretization of \(\varOmega (y,b)=(Cv,v)\) from (H2). Therefore, the estimate (36) follows directly from this second-order condition if only h is taken sufficiently small. \(\square \)

For our further analysis it will be useful to restrict the rhs terms \(\delta \) to a certain neighborhood of \(\delta _{0}=0\). To this aim, for \(\delta =(\delta _{1},\delta _{2},\delta _{3})\) and given \(\rho >0\) define

$$\begin{aligned} \Vert \delta \Vert ^{2}_{D}:= & {} \Vert \delta _{1}\Vert ^{2}_{2}+\Vert \delta _{2}\Vert ^{2}_{(2)} +\Vert \delta _{3}\Vert ^{2}_{2}+ \,\Vert \Delta ^{1}\delta _{3}\Vert ^{2}_{2}, \end{aligned}$$
(37)
$$\begin{aligned} D_{\rho }= & {} \{\delta \in \mathbb {R}^{l}: \; \Vert \delta \Vert _{D}<\,\rho \,\}. \end{aligned}$$
(38)

Lemma 6

Under assumptions (H0)–(H2), there exist constants \(\rho >0\) and \(h'<\bar{h}\) such that, for each \(\delta \in D_{\rho }\) and \(h\le h'\), the variational inequality (LVI’) restricted to \(W_{\beta }^{h}\) has a unique solution \(v^{\delta }\in R^{N}\). On \(D_{\rho }\), the elements \(y^{\delta }\) defined by (30) with \(v=v^{\delta }\) satisfy

$$\begin{aligned} \Vert y^{\delta '}-\,y^{\delta }\Vert _{2}+\,|y^{\delta '}_{N}-y^{\delta }_{N}|\, \le \,c\Vert \delta '-\,\delta \Vert _{D}, \end{aligned}$$
(39)

and the constants \(c,\rho \) are independent of h.  Moreover, there exist unique multipliers \(\nu ^{\delta }_{1,2}\) such that \(\xi ^{\delta }=(z^{\delta },q^{\delta },v^{\delta },\nu ^{\delta })\) (with \((z^{\delta },q^{\delta })\) defined by (25), (26) for \(v=v^{\delta }\)) is a solution of \(\mathrm{(LVI}^{\delta }_{h})\).

Proof

The existence and uniqueness of \(v^{\delta }\) is a consequence of the continuity and strict monotonicity of \(C^{h}\) together with the boundedness of the closed convex set \(W_{\beta }^{h}\) in \(\mathbb {R}^{N}\), resp. the finite-dimensional subspace \(V^{h}\); cf. [27].

It remains to prove the Lipschitz continuity of \(y=y^{\delta }\) w.r.t. \(\delta \): First notice that \(\delta _{3}\) can be interpreted as a piecewise linear function belonging to \(W_{2}^{1}(0,1;\mathbb {R})\). By the Theorem of Morrey [11, Theorem 6.25], on intervals the embedding \(W_{2}^{1}\rightarrow L_{\infty }\) is continuous. Consequently, for \(r^{h}\) the estimate

$$\begin{aligned} \Vert \Delta ^{1}r^{h}(\delta ')-\Delta ^{1}r^{h}(\delta ) \Vert _{2}+\Vert r^{h}(\delta ')-r^{h}(\delta )\Vert _{\infty } \le \,c'\,\Vert \delta '-\delta \Vert _{D} \end{aligned}$$

(with some \(c'>0\) independent of h) follows directly from (29) and the definition of \((z^{h}_{\delta }, q^{h}_{\delta })\). With the abbreviations \(v=v^{\delta }, v'=v^{\delta '}\) etc., from (LVI’) further obtain

$$\begin{aligned} (C^{h}(v'-v),v'-v)\le & {} \left( r^{h}(\delta ')-r^{h}(\delta ),\Delta ^{1}(y'-y)\right) \\= & {} (r^{h}_{N-1}(\delta ')-r_{N-1}^{h}(\delta ))(y'_{N}-y_{N})\\&\quad -\,h\sum _{i=1}^{N-1} \left( \Delta ^{1}(r^{h}(\delta ')-r^{h}(\delta ))\right) _{i-1}(y'_{i}-y_{i})\\\le & {} c\Vert \delta '-\delta \Vert _{D}\left( |y_{N}'-y_{N}|\,+\,\Vert y'-y\Vert _{2}\right) . \end{aligned}$$

Together with (36), the desired Lipschitz property (39) of \(y^{\delta }\) follows. In particular,

$$\begin{aligned} \Vert y^{\delta }\Vert _{2}+\,|y^{\delta }_{N}|\,\le \,c\,\Vert \delta \Vert _{D}. \end{aligned}$$
(40)

Denote \(\phi :=-C^{h}v^{\delta }+\tilde{\sigma }^{h}+\,r^{h}(\delta )\in \,\mathbb {R}^{N}\). Without loss of generality, one can write \(\phi =\phi ^{1}-\,\phi ^{2}\) for some \(\phi ^{1},\phi ^{2}\ge \,0\). From (LVI’) the following relation is obtained:

$$\begin{aligned} (\phi ,v-v^{\delta })= (\phi ^{1},w-w^{\delta })-(\phi ^{2},w-w^{\delta })\le 0 \end{aligned}$$
(41)

for \(w^{\delta }=v^{\delta }+\tilde{u}^{h}\) and all \(w\in \tilde{u}^{h}+ W^{h}_{\beta }\). Using \(\phi ^{1,2}\) as candidates for \(\nu _{1,2}^{\delta }\), it follows directly from (28) that the only possible choice is given by

$$\begin{aligned} \phi ^{1}=\,[\phi ]_{+},\qquad \phi ^{2}=\,[\phi ]_{-}. \end{aligned}$$

Then, by (41) we have  \((\phi ^{1},w-w^{\delta })\le 0, (\phi ^{2},w-w^{\delta }) \ge 0 \) for all \(w\in \tilde{u}^{h}+\,W^{h}_{\beta }\). Consequently, for all i with \(t_{i}\not \in J_{\beta }\) obtain

$$\begin{aligned} \phi _{i}=[\phi _{i}]_{+}>0 \;\,\Rightarrow \;\, w^{\delta }_{i}=1,\qquad \phi _{i}=[\phi _{i}]_{-}>0 \;\,\Rightarrow \;\, w^{\delta }_{i}=-1. \end{aligned}$$

In case \(t_{i}\in J_{\beta }\) remember

$$\begin{aligned} \phi _{i}\,=\,B^{T}(\tilde{S}y+\tilde{W}b)_{i} +\,\tilde{\sigma }^{h}_{i}+\,r^{h}_{i}(\delta ). \end{aligned}$$

Inserting \(y=y^{\delta }, b=y^{\delta }_{N}\), it follows that

$$\begin{aligned} |\phi _{i}-\tilde{\sigma }^{h}_{i}|\,\le \,c(\Vert y^{\delta }\Vert _{2}+|y^{\delta }_{N}|+\Vert r^{h}(\delta )\Vert _{\infty }) =\text{ O }(\Vert \delta \Vert _{D}); \end{aligned}$$

cf. (40). Choosing \(\rho \) and \(h'\) sufficiently small one can guarantee \(|\phi _{i}|>\beta /2>0\) uniformly for \(\delta \in D_{\rho },\,h<h'\) and \(t_{i}\in J_{\beta }\) so that \(\phi \) has the same sign as \(\sigma ^{0}\). Hence, (27), (28) are fulfilled if and only if \(\nu ^{\delta }_{1}=[\phi ]_{+}, \,\nu ^{\delta }_{2}=[\phi ]_{-}\). \(\square \)

Lemma 7

Under Assumptions (H0)–(H2), there exist constants \(h', \rho \) and \(\rho '\) such that, for all \(h<h'\) and \(\delta \in D_{\rho '}\), the following statements hold:

  1. (i)

     On the set  \(\varXi _{\rho }=\{\xi =(z,q,v,\nu ):\, v=\Delta ^{1}y,\,y_{0}=0,\; \Vert y\Vert ^{2}_{2}+\,|y_{N}|^{2}<\rho ^{2}\}\), the variational inequality (\(\hbox {LVI}^{h}_{\delta }\)) has a unique solution \(\xi ^{h}(\delta )\),

  2. (ii)

    the components \((z^{h}(\delta ), q^{h}(\delta ), y^{h}(\delta ), \nu ^{h}(\delta ))\) depend Lipschitz continuously on \(\delta \) in the sense

    $$\begin{aligned} \begin{array}{c} \max \{\Vert z^{\delta }-\,z^{\delta '}\Vert _{2},\,\Vert q^{\delta }-\,q^{\delta '}\Vert _{2},\, \Vert y^{\delta }-\,y^{\delta '}\Vert _{2},\, \Vert \nu ^{\delta }-\,\nu ^{\delta '}\Vert _{\infty }\} \le \,c\,\Vert \delta -\delta '\Vert _{D},\\ |\pi z^{\delta }-\,\pi z^{\delta '}|\,+\,|\pi q^{\delta }-\,\pi q^{\delta '}|\,\le \,c\,\Vert \delta -\delta '\Vert _{D} \end{array} \end{aligned}$$

    where \(\pi :\mathbb {R}^{n(N+1)}\rightarrow \mathbb {R}^{2n}\) denotes the boundary trace operator \(\pi \phi =(\phi _{0},\phi _{N})\). The constant c herein does not depend on h or \(\delta , \delta '\in D_{\rho '}\).

Proof

For \(h<h'\) and \(\Vert \delta \Vert _{D}<\rho '\) with appropriately chosen \(\rho '\), it follows from Lemma 6 that \((\hbox {LVI}^{\delta }_{h}\)) has a solution \(\xi =\xi ^{\delta }\). Due to (40), one can always take \(\rho '\) small enough to ensure \(\xi \in \varXi _{\rho }\) for given \(\rho >0\). Thus, part (i) of the proof reduces to verifying the solution uniqueness on the set \(\varXi _{\rho }\):

Assume for the moment, that (\(\hbox {LVI}^{\delta }_{h}\)) has a second solution \(\hat{\xi }^{\delta }\not =\xi ^{\delta }\), and consider

$$\begin{aligned} \hat{\phi }\,=\,-\hat{C}^{h}\hat{y}^{\delta }+\, \tilde{\sigma }^{h}+r^{h}(\delta ). \end{aligned}$$

By (35) and standard estimates for (32), (33), obtain

$$\begin{aligned} \Vert \hat{C}^{h}\hat{y}^{\delta }\Vert _{\infty } \le \,c\left( \Vert \hat{y}^{\delta }\Vert _{2}+\,|\hat{y}^{\delta }_{N}|\right) \, =\;\text{ O }(\rho ). \end{aligned}$$

On the other hand, (29) yields

$$\begin{aligned} \Vert r^{h}(\delta )\Vert _{\infty }\le \,c\Vert \delta \Vert _{D}\,=\;\text{ O }(\rho '). \end{aligned}$$

If now \(\rho ,\rho '\) and \(h'\) are sufficiently small then, in particular, \(\Vert \hat{\phi }-\tilde{\sigma }^{h}\Vert _{\infty }<\beta /4\) and thus, for \(\hat{\nu }^{\delta }_{1}=[\hat{\phi }]_{+},\, \hat{\nu }^{\delta }_{2}=[\hat{\phi }]_{-}\) deduce

$$\begin{aligned} \Vert \hat{\nu }^{\delta }_{1}-\tilde{\mu }^{h}_{1}\Vert _{\infty }+\, \Vert \hat{\nu }^{\delta }_{2}-\tilde{\mu }^{h}_{2}\Vert _{\infty }<\,\beta /2. \end{aligned}$$

Lemma 4 therefore shows that \(\hat{v}^{\delta }\) solves (LVI’) on \(W_{\beta }^{h}\) and hence, coincides with \(v^{\delta }\). By construction of \(\hat{z}^{\delta },\hat{q}^{\delta }\) and \(\hat{\nu }^{\delta }\) according to (\(\hbox {LVI}^{\delta }_{h}\)) it follows that \(\hat{\xi }^{\delta }=\,\xi ^{\delta }\), i.e., for \(\delta \in D_{\rho '}\) the solution is unique on \(\varXi _{\rho }\).

Part (ii): The estimates are direct consequences of (39), Lemma 6, together with the construction

$$\begin{aligned} \zeta= & {} Sy,\qquad z\,=\,\zeta \,+\,By,\qquad q\,=\,\tilde{S}y\,+\,\tilde{W}y_{N},\\ \nu _{1}= & {} [\hat{C}^{h}y+\,r^{h}(\delta )]_{+},\quad \nu _{2}\,=\,[\hat{C}^{h}y+\,r^{h}(\delta )]_{-} , \end{aligned}$$

and formula (29) for \(r^{h}(\delta )\). \(\square \)

4.3 Strong regularity

For variational inequalities of type (20), a fundamental existence and Lipschitz stability result was given by Robinson in [42]. The crucial characterizing property concerns in the strong regularity which, for continuously Fréchet differentiable mappings \(\psi \) from certain Banach space \(\varXi \) to its dual space, allows to prove existence and local uniqueness of solution, and further its Lipschitz continuous dependence on problem parameters [42, Theorem 2.1]:

Definition 1

[42, Definition 1]  Let \(\varXi \) be a normed linear space, and let \(\varOmega \) be an open subset of \(\varXi \) containing a point \(\bar{\xi }\). Let C be a closed convex set in \(\varXi \), and let \(\psi :\;\varOmega \rightarrow \,\varXi ^{*}\) (where \(\varXi ^{*}\) is the topological dual of \(\varXi \)) be Fréchet differentiable at \(\bar{\xi }\). Suppose that

$$\begin{aligned} 0\;\in \;\psi (\xi )\,+\,N_{C}(\xi ) \end{aligned}$$
(42)

has \(\bar{\xi }\) as a solution, and define, for \(\xi \in \varXi \), 

$$\begin{aligned} T_{C}(\xi ):=\,\psi (\bar{\xi })\,+\,\psi '(\bar{\xi })(\xi -\bar{\xi }) \,+\,N_{C}(\xi ). \end{aligned}$$

We say that (42) is strongly regular at \(\bar{\xi }\) with associated Lipschitz constant \(\lambda \) if there exist neighborhoods U of the origin in \(\varXi ^{*}, V\) of \(\bar{\xi }\) in \(\varXi \) such that the restriction to U of \(T_{C}^{-1}\cap V\) is a single-valued function from U to V which is Lipschitzian on U with modulus \(\lambda \).

Remark 1

A closer look on the main steps of the proof in [42] shows that, without loss of generality, the neighborhood U of rhs perturbations \(\bar{\delta }\) for the linearized problem (23) can be restricted to the subspace \(\varXi ^{0}\subset \varXi ^{*}\) where all \(\bar{\delta }\) components corresponding to linear components of \(\psi \) vanish. In the forthcoming, we will call (42) also strongly regular (in the sense of Robinson) if the definition holds in this slightly relaxed form.

In order to make usage of the strong regularity approach for the variational inequalities (\(\hbox {VI}_{\delta }^{h}\)), the space \(\mathbb {R}^{d}\) will be equipped with a norm reflecting the use of Goh’s transformation in deriving (39) and the estimates in Lemma 7: Define \(\varXi =(\mathbb {R}^{d}, |\Vert \cdot \Vert |)\) with the norm of an element \(\xi =(p,x,u,\mu )\) given by

$$\begin{aligned} |\Vert \xi \Vert |\,:=\,\left[ \Vert p\Vert _{(2)}^{2}+\Vert x\Vert _{(2)}^{2}+\Vert u\Vert _{(-1)}^{2}+\Vert \mu \Vert _{2}^{2}\right] ^{1/2}, \end{aligned}$$
(43)

where

$$\begin{aligned} \Vert u\Vert _{(-1)}:=\,\left( y_{N}^{2}+h\sum _{i=1}^{N}y_{i}^{2}\right) ^{1/2},\qquad y_{i}=\,h\sum _{k=0}^{i-1}u_{k}. \end{aligned}$$

Then \(\varXi \) is a Banach space and its dual \(\varXi ^{*}\) can be described as \((\mathbb {R}^{d},|\Vert \cdot \Vert |_{*})\) with

$$\begin{aligned} |\Vert \varTheta \Vert |_{*}:=\,\left[ \Vert \phi \Vert _{(2)}^{2}+\Vert \rho \Vert _{(2)}^{2}+\Vert \Delta ^{1}\vartheta \Vert _{2}^{2} +\vartheta _{N}^{2}+\Vert \omega \Vert _{2}^{2}\right] ^{1/2} \end{aligned}$$
(44)

for \(\varTheta =(\phi , \rho ,\vartheta ,\omega )\in \mathbb {R}^{d}\).  Indeed, one can transform \((\vartheta ,u)\) as follows

$$\begin{aligned} (\vartheta ,u)= & {} h\sum _{i=0} ^{N-1}\vartheta _{i}u_{i}\,=\; h\sum _{i=0} ^{N-1}\vartheta _{i}(\Delta ^{1}y)_{i}\\= & {} -h\sum _{i=1} ^{N}y_{i}(\Delta ^{1}\vartheta )_{i-1}\,+\;\vartheta _{N}y_{N}, \end{aligned}$$

and consequently, \(|(\vartheta ,u)|\le \Vert u\Vert _{(-1)}(\Vert \Delta ^{1}\vartheta \Vert ^{2}_{2}+\,| \vartheta _{N}|^{2})^{1/2}\) holds true. Notice that the last estimate is sharp whenever \(\vartheta \) solves

$$\begin{aligned} (\Delta ^{1}\vartheta )_{i-1}=\,y_{i}, \;\;i=1,\dots ,N,\quad \vartheta _{N}=y_{N}. \end{aligned}$$

Moreover, on the subspace \(\varXi ^{0}\) of \(\varXi ^{*}\) with zero \(\varTheta \)-entries \(\phi _{N}=0\) and \(\omega =0\) corresponding to linear parts of \(\psi \), the norm \(|\Vert \bar{\delta }\Vert |_{*}\) is equivalent to \(\Vert \delta \Vert _{D}\) from (37), and the constants in the related estimates do not depend on h.

Remark 2

It is well known that all norms in finite-dimensional space are equivalent. In particular, for the construction (43) the following estimates hold:

$$\begin{aligned} \Vert u\Vert _{(-1)}^{2}=y_{N}^{2}+h\sum _{i=1}^{N}y_{i}^{2}\le \, 2\,\Vert y\Vert _{\infty }^{2}\le \;2\, \Vert u\Vert _{2}^{2} \end{aligned}$$

(cf. Lemma 14, Appendix);  and  

$$\begin{aligned} \Vert u\Vert _{2}^{2}=h^{-1}\sum _{i=0}^{N-1}(y_{i+1}-y_{i})^{2}\le 4h^{-1}\sum _{i=1}^{N}y_{i}^{2}\le 4h^{-2}\Vert u\Vert _{(-1)}^{2}. \end{aligned}$$

Thus, for each fixed \(h>0\), the norm \(|\Vert \cdot \Vert |\) is equivalent to the Euclidean norm in \(\mathbb {R}^{d}\) but the related constants are not independent of h.

Lemma 8

Under assumptions (H0)–(H2), there exists a constant \(h'\) such that each variational inequality \((\mathrm{VI}_{\delta }^{h})\), \(h<h'\), is strongly regular (in the sense of Robinson) for \(\delta =\tilde{\delta }^{h}\) at the solution \(\tilde{\xi }^{h}\), and the related neighborhoods \(U=\{\bar{\delta }\in \varXi ^{0}:\; \Vert \delta -\tilde{\delta }^{h}\Vert _{D}\le \,\rho '\}, V=\{\xi \in \varXi :\; |\Vert \xi -\tilde{\xi }^{h}\Vert |<\,\bar{\rho }\}\) and the Lipschitz modulus \(\lambda \) can be chosen independently of h.

Proof

With the definitions of \(\varXi , \varXi ^{*}\) and related norms \( |\Vert \cdot \Vert |\) and \( |\Vert \cdot \Vert |_{*}\) from (43), (44), the assertion is deduced for \((\psi -\delta )\) from Lemmas 6, 7 and the estimates for \(\Vert \tilde{\delta }^{h}\Vert _{D}\) from Lemma 2: with the constants \(\rho , \rho ', c\) from Lemma 7, sufficiently small \(h'\) and appropriately chosen \(\bar{\rho }=\hbox {O}(\rho )\), the Lipschitz continuity follows with a modulus \(\lambda =\hbox {O}(c)\). \(\square \)

In order to verify local uniqueness and Lipschitz behavior of solutions in (\(\hbox {VI}^{h}_{\delta }\)), first refer to [42, Theorem2.1]: indeed, assumption (H0) together with Lemma 1 guarantees solution existence for each of the problems (\(\hbox {VI}^{h}_{\delta }\)). Further, for arbitrary \(\rho _{0}>0\) and \(\bar{D}_{0}=\{\bar{\delta }\in \varXi ^{0}:\,\Vert \delta \Vert _{D}\le \rho _{0}\}, \psi (\bar{\delta },\xi ):=\,\psi (\xi )-\bar{\delta }\) as a mapping from \(\bar{D}_{{0}}\times \varXi \) to \(\varXi ^{*}\) is twice continuously differentiable so that, for fixed h, the assumptions of Robinson’s Inverse Function Theorem are fulfilled. However, for obtaining uniform w.r.t. h estimates of related neighborhoods and Lipschitz moduli, the fixpoint approach used in [42] has to reconsidered.

4.4 Lipschitz stability result

The inclusion (\(\hbox {VI}^{h}_{\delta }\)) can be transformed to a fixpoint problem as follows:  for given \((\delta ,u)\in \mathbb {R}^l\times \mathbb {R}^N\) with \(\Vert u\Vert _\infty \le 1\), denote by \((x,p)=(x(\delta ,u),p(\delta ,u))\) the solution of (16), (17). Further, define \(\mu =\mu (\delta ,u)\) by \(\sigma =B^Tp(\delta ,u)-\delta _3\) and \(\mu _1=[\sigma ]_+, \,\mu _2=[\sigma ]_-\),  and introduce

$$\begin{aligned} r(\delta ,u)\,:=\,T(\xi (\delta ,u))-\psi (\xi (\delta ,u))+\bar{\delta } \end{aligned}$$
(45)

with T given by (24).  Obviously, \(\xi =\xi (\delta ,u)\) solves (\(\hbox {VI}^{h}_{\delta }\)) if and only if

$$\begin{aligned} r(\delta ,u)\in T(\xi (\delta ,u))+N_C(\xi (\delta ,u))=:T_C(\xi (\delta ,u)). \end{aligned}$$

By Lemma 6, for \(\delta \in D_\rho \) the set \(\varLambda (\delta )\,:=\,T_C^{-1}(\bar{\delta })\) is nonempty. If \(\varLambda _u(\delta )\) denotes the set of u-components related to \(\xi \in \varLambda (\delta )\), and

$$\begin{aligned} \varPhi _\delta (u):=\;\varLambda _u(r(\delta ,u)), \end{aligned}$$
(46)

then the element \(\xi =\xi (\delta ,u)\) solves (VI\(^{h}_{\delta }\))  if and only if  \(u\,\in \,\varPhi _\delta (u)\)  (see also [15, Lemma 2.1]). This fixpoint characterization allows to prove

Theorem 1

Under assumptions (H0)–(H2), there exist constants \(h', \bar{\rho }'\) and \(\epsilon \) such that, for all \(h<h'\) and \(\delta \in D'=\{\delta \in \mathbb {R}^{l}:\;\Vert \delta -\tilde{\delta }^{h}\Vert _{D}<\bar{\rho }'\}\), problem \((\mathrm{VI}_{\delta }^{h}\)) has a unique solution \(\xi ^{h}_{\delta }\) on the set   \(V_{\epsilon }=\,\{\xi \in \mathbb {R}^{d}:\,\Vert u-\tilde{u}^{h}\Vert _{(-1)}\le \,\epsilon \}\) and, as an element of \(\varXi ,~\xi ^{h}_{\delta }\) depends Lipschitz continuously on \(\bar{\delta }\in \varXi ^{*}\). The associated Lipschitz modulus as well as the radii \(\epsilon , \bar{\rho }'\) are independent of  h.

Before proving the theorem, some auxiliary estimates will be provided.

First notice that, under assumption (H0), the solutions \((x(u,\delta ), p(u,\delta ))\) of (16), (17) are uniformly bounded, cf. Lemma 1:  if \(\Vert \delta \Vert _{D}< \rho _{0}\) then

$$\begin{aligned} \Vert x\Vert _{\infty }+\Vert \Delta ^{1}x\Vert _{2}+ \Vert p\Vert _{\infty }+\Vert \Delta ^{1}p\Vert _{2} \,\le \, M_{0} \end{aligned}$$
(47)

for some constant \(M_{0}\) independent of \(u, \delta \) and h. Shrinking \(h'\) if necessary, similar bounds are obtained for \((\tilde{x}^{h}, \tilde{p}^{h})\) by definition of these functions and the estimates for \(\tilde{\delta }^{h}\) from Lemma 2. Therefore,

$$\begin{aligned} \Vert z'\Vert _{\infty }+\Vert \Delta ^{1}z'\Vert _{2}+ \Vert q'\Vert _{\infty }+\Vert \Delta ^{1}q'\Vert _{2} \,\le \, M_{1} \end{aligned}$$
(48)

holds true for \(z'=x-\tilde{x}^{h}, q'=p-\tilde{p}^{h}\) and \(M_{1}:=2M_{0}\). As long as \(h<h'\) and \(\Vert \delta \Vert _{D}<\rho _{0}\), Lemma 13 from Appendix further yields

$$\begin{aligned} \Vert z'\Vert ^{2}_{\infty }+\, \Vert q'\Vert ^{2}_{\infty } \,\le \, M_{2}\left( h'+\,\Vert z'\Vert _{(2)}+\,\Vert q'\Vert _{(2)}\right) \end{aligned}$$
(49)

with a constant \(M_{2}\) independent of \(u, \delta \) and h.

Lemma 9

Let \(\rho >0\) be arbitrarily given. If \(\bar{\rho }'\) and \(\epsilon \) are suffiently small, \(h<h'\) and \(\delta \in D'\), then every solution of (\(\hbox {VI}_{\delta }^{h}\)) in \(V_{\epsilon }\) satisfies  \(|\Vert \xi -\tilde{\xi }^{h}\Vert |<\rho \).

Proof

If \(\xi =(p,x,u,\mu )\) is a solution of (\(\hbox {VI}_{\delta }^{h}\)) then \(\xi =\xi (\delta ,u)\). By \(y, \tilde{y}^{h}\) denote the integrated controls \(y_{i}=h\sum _{k=0}^{i-1}v_{i}\) related to \(v=u\) or \(v=\tilde{u}^{h}\) resp., and set \(\eta :=x-By, \,\tilde{\eta }^{h}=\tilde{x}^{h}-B\tilde{y}^{h}\). For \(z:=x-\tilde{x}^{h}, \,q:=p-\tilde{p}^{h}, \zeta :=\eta -\tilde{\eta }^{h}\) and \(\omega :=y-\tilde{y}^{h}\), under (H0) we conclude

$$\begin{aligned} (\Delta ^{1}\zeta )_{i}= & {} f(\eta _{i}+By_{i})-f(\tilde{\eta }_{i}^{h}+B\tilde{y}_{i}^{h}) \,+\,\delta _{1i}-\tilde{\delta }^{h}_{1i},\\ |(\Delta ^{1}\zeta )_{i}|\le & {} L'(|\zeta _{i}|+|B|\,|\omega _{i}|) \,+\,|\delta _{1i}-\tilde{\delta }^{h}_{1i}|,\\ |(\Delta ^{1}q)_{i}|\le & {} L'(|q_{i+1}|+|z_{i+1}|) \,+\,|\delta _{2i}-\tilde{\delta }^{h}_{2i}|, \end{aligned}$$

where \(L'\) is the common Lipschitz modulus of f and \(H_{x}\) on the set \(\{(x,p): \;\Vert x\Vert _{\infty }+\,\Vert p\Vert _{\infty }\le \,M_{0}\}\); cf. (47). Applying Lemma 14 from Appendix to the equation for \(\zeta \) yields

$$\begin{aligned} \Vert \zeta \Vert _{\infty }\le \,c(\Vert \delta -\tilde{\delta }^{h}\Vert _{D}+\Vert \omega \Vert _{2}) \end{aligned}$$

with some constant c independent of \(h,\delta \) and u.

Consequently,  \(\Vert z\Vert _{2}\le (c+|B|)(\Vert \delta -\tilde{\delta }^{h}\Vert _{D}+\Vert \omega \Vert _{2})\).  Taking into account this relation and applying Lemma 14 to q now, an analogous estimate for \(\Vert q\Vert _{\infty }\) is obtained. Summing up, the inequality

$$\begin{aligned} \Vert x-\tilde{x}^{h}\Vert _{2}+\Vert p-\tilde{p}^{h}\Vert _{\infty }+\Vert \mu -\tilde{\mu }^{h}\Vert _{\infty } \le \,\text{ O }(\Vert \delta -\tilde{\delta }^{h}\Vert _{D}+\,\Vert u-\tilde{u}^{h}\Vert _{(-1)}) \quad \end{aligned}$$
(50)

follows and hence the assertion. \(\square \)

The preparation for proving Theorem 1 will be continued with an analysis of  \(r=r(\delta ,u)\):  by construction, \(r(\delta ,u)\!=\!(r_{1},r_{2},r_{3},r_{4})\in \varXi ^{0}\)  (i.e. \(r_{1N}, r_{4}\) vanish), and \(r_{3}=\delta _{3}-\tilde{\delta }^{h}_{3}\). Further,

$$\begin{aligned} r_{1i}= & {} f(x_{i})-f(x^{0}_{i})-A_{i}(x_{i}-x_{i}^{0})+\, \delta _{1i}-\tilde{\delta }^{h}_{1i},\\ r_{2i}= & {} (\nabla f(x^{0}_{i+1})-\nabla f(x_{i+1}))^{T}p_{i+1}\! -\!\left. \nabla ^{2}_{x}(\nabla f(x^{0})^{T}p^{0})\right| _{t_{i+1}}\!\!(x_{i+1}-x_{i+1}^{0}) \\&\quad +\;\delta _{2i}-\tilde{\delta }^{h}_{2i},\\ r_{2N}= & {} \nabla k(x^{0}_{N})-\nabla k(x_{N})\,+\,K(x_{N}-x_{N}^{0})+ \delta _{2N}-\tilde{\delta }^{h}_{2N}. \end{aligned}$$

Consequently, from (48) and

$$\begin{aligned} |r_{1i}|\le & {} c\Vert z'\Vert _{\infty }|x_{i}-x^{0}_{i}|+|\delta _{1i}-\tilde{\delta }^{h}_{1i}|\le cM_{1}|x_{i}-x^{0}_{i}|+|\delta _{1i}-\tilde{\delta }^{h}_{1i}|, \quad i\le N-1,\\ |r_{2i}|\le & {} c(\Vert z'\Vert _{\infty }+\,\Vert q'\Vert _{\infty }) \cdot (|x_{i+1}-x^{0}_{i+1}|+|p_{i+1}-p^{0}_{i+1}|)+ |\delta _{2i}-\tilde{\delta }^{h}_{2i}|,\;\;\; i\le N, \end{aligned}$$

it is deduced that, for some constant \(c_{r}\) independent of \(u,h,\delta \),

$$\begin{aligned} |\Vert r\Vert |_{*}\le & {} {c}\left( \Vert z'\Vert _{\infty }+\,\Vert q'\Vert _{\infty }\right) \cdot (\Vert x-\tilde{x}^{h}\Vert _{(2)}+\Vert p-\tilde{p}^{h}\Vert _{(2)}) \,+\,\Vert \delta -\tilde{\delta }^{h}\Vert _{D}\end{aligned}$$
(51)
$$\begin{aligned}\le & {} {c}_{r}\rho \,+\,\bar{\rho }' \end{aligned}$$
(52)

if  \(h<h',\;\delta \in D',\; |\Vert \xi -\tilde{\xi }^{h}\Vert |\le {\rho }\), and \(\bar{\rho }'<\rho _{0}\).

Finally consider \(r^{1}=r(\delta ,u^{(1)})\) and \(r^{2}=r(\delta ,u^{(2)})\) for given \(u^{(1)}, u^{(2)}\) and fixed \(\delta \in D'\). Denoting by \((x^{(j)},p^{(j)})\) the solution of (16), (17), and setting \(z^{j}=x^{(j)}-\tilde{x}^{h}, q^{j}=p^{(j)}-\tilde{p}^{h}\) for \(j=1,2\), in analogy to (51) obtain

$$\begin{aligned} |\Vert r^{1}-r^{2}\Vert |_{*}\,\le \, {c}\max _{j=1,2}\{\Vert z^{j}\Vert _{\infty }+\Vert q^{j}\Vert _{\infty } \} \cdot (\Vert x^{(1)}-{x}^{(2)}\Vert _{(2)}+\Vert p^{(1)}-{p}^{(2)}\Vert _{(2)}). \end{aligned}$$
(53)

For the elements \(z^{j}, q^{j},\;j=1,2\), (49) can be applied so that, with a constant \(\rho \) such that \(|\Vert \xi ^{(j)}-\tilde{\xi }^{h}\Vert |<\rho \) for \(j=1,2\), the auxiliary result

$$\begin{aligned} |\Vert r^{1}-r^{2}\Vert |_{*}\,\le \,\text{ O }(\sqrt{h'}+\sqrt{\rho }) \cdot (\Vert x^{(1)}-{x}^{(2)}\Vert _{(2)}+\Vert p^{(1)}-{p}^{(2)}\Vert _{(2)}). \end{aligned}$$

is obtained. If we repeat the arguments from the proof of (50), Lemma 9, we deduce  \(\Vert x^{(1)}-{x}^{(2)}\Vert _{(2)}+\Vert p^{(1)}-{p}^{(2)}\Vert _{(2)}\le \,c\Vert u^{(1)}-u^{(2)}\Vert _{(-1)}\) and thus, end up with

$$\begin{aligned} |\Vert r^{1}-r^{2}\Vert |_{*}\, \le \,c_{r}'(\sqrt{h'}+\sqrt{\rho })\cdot \Vert u^{(1)}-u^{(2)}\Vert _{(-1)} \end{aligned}$$
(54)

where the constant \(c_{r}'\) is independent of \(h,\delta , u^{(1)}\) and \(u^{(2)}\) as long as \(\xi ^{(1)},\xi ^{(2)}\) satisfy \(\max \{|\Vert \xi ^{(1)}-\tilde{\xi }^{h}\Vert |,|\Vert \xi ^{(2)}-\tilde{\xi }^{h}\Vert |\}<\rho \).

Proof of Theorem 1

Define  \(W_{\epsilon }=\{w\in \mathbb {R}^{N}:\; \Vert w-\tilde{u}^{h}\Vert _{(-1)}\le \epsilon \}\). According to Lemma 9, one can choose \(\epsilon \) and \(\bar{\rho }'\) small enough to ensure

$$\begin{aligned} |\Vert \xi (\delta ,w)-\tilde{\xi }^{h}\Vert |\,\le \,\rho \,\le \,\bar{\rho }\, \end{aligned}$$

with the constant \(\bar{\rho }\) from Lemma 8 for all \(w\in W_{\epsilon }\) and arbitrary \(h<h'\). Shrinking \(\epsilon ,\,h'\) and \(\bar{\rho }'\) if necessary and taking into account (52) as well as Lemma 2, the estimate

$$\begin{aligned} |\Vert r(\delta ,w)-\bar{\delta }^{h}\Vert |_{*}\le \,{c}_{r}{\rho }\,+\,\bar{\rho }' +\,\Vert \tilde{\delta }^{h}\Vert _{D}<\,\rho ' \end{aligned}$$

with constant \(\rho '\) from Lemma 8 will hold uniformly for \(w\in W_{\epsilon }\). Thus, for arbitrary \(\delta \in D'\) and \(h<h'\),  the mapping \(\varPhi _{\delta }(w)\) from (46) is well-defined and Lipschitz continuous on \(W_{\epsilon }\): it is sufficient to notice that the underlying map \(\varLambda \) has the Lipschitz modulus \(\lambda \).

Let \(w^{1},w^{2}\) be any two points from \(W_{\epsilon }\). Using once more Lemma 8 we have

$$\begin{aligned} \left\| \varPhi _{\delta }(w^{1})-\varPhi _{\delta }(w^{2})\right\| _{(-1)}\le & {} |\Vert \varLambda (r(\delta ,w^{1}))-\varLambda (r(\delta ,w^{2}))\Vert |\\\le & {} \lambda \,|\Vert r(\delta ,w^{1})-r(\delta ,w^{2})\Vert |_{*}\\\le & {} \lambda c'_{r}(\sqrt{h'}+\sqrt{\rho })\cdot \Vert w^{1}-w^{2}\Vert _{(-1)} \end{aligned}$$

and \(\rho =\hbox {O}(\epsilon +\bar{\rho }')\);  see (53), (50) and Lemma 13. Shrinking \(h', \epsilon \) and \(\bar{\rho }'\) if necessary, one can make the last term satisfy

$$\begin{aligned} \left\| \varPhi _{\delta }(w^{1})-\varPhi _{\delta }(w^{2})\right\| _{(-1)}\,\le \, \kappa \, \Vert w^{1}-w^{2}\Vert _{(-1)} \end{aligned}$$
(55)

with a constant \(\kappa <1\). Hence \(\varPhi _{\delta }\) is strictly contractive.

Next we will show that \(\varPhi _{\delta }\) is a self-map of \(W_{\epsilon }\):

First, consider \(w^{0}=\tilde{w}^{h}, \,\delta _{0}=\tilde{\delta }^{h}\): obviously,   \(w^{0}=\varPhi _{\delta _{0}}(w^{0})\) and

$$\begin{aligned} \left\| \varPhi _{\delta }(w^{0})-w^{0}\right\| _{(-1)}\le & {} |\Vert \varLambda (r(\delta ,w^{0}))-\varLambda (r(\delta _{0},w^{0}))\Vert |\\\le & {} \lambda \,|\Vert r(\delta ,w^{0})-r(\delta _{0},w^{0})\Vert |_{*}\\\le & {} \lambda \,\Vert \delta -\delta _{0}\Vert _{D}\le \,\lambda \,\bar{\rho }' \end{aligned}$$

follows by the definition of r, and from (50), Lemma 9. Further, for arbitrary \(w\in W_{\epsilon }\), from the contractivity of \(\varPhi _{\delta }\) conclude

$$\begin{aligned} \left\| \varPhi _{\delta }(w)-w^{0}\right\| _{(-1)}\le & {} \left\| \varPhi _{\delta }(w)-\varPhi _{\delta }(w^{0})\right\| _{(-1)} \,+\,\left\| \varPhi _{\delta }(w^{0})-w^{0}\right\| _{(-1)}\\\le & {} \kappa \,\epsilon \,+\,\lambda \,\bar{\rho }'. \end{aligned}$$

The latter can be made smaller than \(\epsilon \) : indeed, the above construction shows \(\kappa =\hbox {O}(\sqrt{h'}+\sqrt{\epsilon }+\sqrt{\bar{\rho }'})\) so that   \(\lambda \bar{\rho }'\) can be made smaller than \((1-\kappa )\epsilon \) by shrinking \(\bar{\rho }'\) if necessary.

The properties of \(\varPhi _{\delta }:\;W_{\epsilon } \rightarrow W_{\epsilon } \) guarantee the existence of a fixpoint \(w=u^h_\delta \) in \(W_{\epsilon }\) corresponding to the unique solution \(\xi ^h_\delta =\xi (\delta ,u^h_\delta )\) of (\(\hbox {VI}^{h}_{\delta }\)) in \(V_\epsilon \). The Lipschitz continuity of \(u^h_\delta \) w.r.t. \({\delta }\) is obtained similarly to the proof in [42, Theorem 2.1] with Lipschitz modulus \(\lambda '=\lambda \,(1-\kappa )^{-1}\). Together with condition (50), this fact yields the Lipschitz continuous dependence of \(\xi ^h_\delta \) on \({\delta }\) with a modulus independent of h and thus, completes the proof. \(\square \)

5 Control properties

In this section, consider (\(\hbox {VI}_{\delta }^{h}\)) in the special case that \(\delta =0\). As it was shown in the previous section, there exists a unique solution \(\xi _{0}^{h}=\hat{\xi }=(\hat{p},\hat{x},\hat{u},\hat{\mu })\) of the problem close to \(\tilde{\xi }^{h}\) provided h is sufficiently small. Due to (18), (19),  \(\hat{\sigma }:=\,B^{T}\hat{p}\) is the discrete switching function for \(\hat{u}\): with \(\hat{\mu }_{1}=[\hat{\sigma }]_{+}, \hat{\mu }_{2}=[\hat{\sigma }]_{-}\) we have

$$\begin{aligned} -1\le \hat{u}_{i}\le 1,\qquad (\hat{u}_{i}-1)[\hat{\sigma }_{i}]_{+}= (\hat{u}_{i}+1)[\hat{\sigma }_{i}]_{-}=0 \end{aligned}$$

for \(i=0,\dots ,N-1\).

In [14], for continuous problems of type (VI) depending on a real parameter it was shown that (H1) together with the assumptions

$$\begin{aligned} \begin{array}{c} R\,=\,-p^{T}\left[ B,[f^0,B]\right] \,=-B^TH^0_{xx}B\,\ge \,\bar{m}\,>\,0\qquad \text{ on }\,\,I_{\beta },\\ \min \{|\ddot{\sigma }^{0}(\tau _{s}-0)|, |\ddot{\sigma }^{0}(\tau '_{s}+0)|\} \,\ge \,\bar{m}\,>\,0\qquad \end{array} \end{aligned}$$

(see [14, Assumptions 2.2-3]), ensures the bang-singular-bang control structure to be stable under parameter perturbation. Similar results will be proved now for the discretized problems (\(\hbox {VI}_{0}^{h}\)) for sufficiently small step size parameter h. To this aim, the discrete solution \((\hat{u},\hat{\sigma })\) will be compared to the reference data \((u^{0},\sigma ^{0})\) from (VI). For simplicity, the discrete functions \((\tilde{u},\tilde{\sigma })\in \mathbb {R}^{N}\times \mathbb {R}^{N}\) defined by \(\tilde{u}_{i}=u^{0}(t_{i}+0),\,\tilde{\sigma }_{i}=\sigma ^{0}(t_{i})\) will again be denoted by \((u^{0},\sigma ^{0})\).

Lemma 10

For the switching function \(\hat{\sigma }\), the following estimate holds:

$$\begin{aligned} \Vert \hat{\sigma }-\sigma ^{0}\Vert _{\infty }+ \,\Vert \Delta ^{1}(\hat{\sigma }-\sigma ^{0})\Vert _{2} =\text{ O }(h). \end{aligned}$$

Moreover, \(\hat{\sigma }_{i}\,=\,\hat{P}_{i}\,+\,\hat{R}_{i}\,\hat{u}_{i}\)   where \(\hat{P},\,\hat{R}\) satisfy

$$\begin{aligned} \begin{array}{l} \Vert \hat{P}-P\Vert _{2}+\,\Vert \hat{R}-R\Vert _{2}=\text{ O }(h),\\ \Vert \Delta ^{1}(\hat{\sigma }-\sigma ^{0})\Vert _{\infty }+\,\Vert \hat{P}-P\Vert _{\infty }+\,\Vert \hat{R}-R\Vert _{\infty }= \text{ O }(\Vert \hat{x}-x^{0}\Vert _{\infty }+\,h) \end{array} \end{aligned}$$

(The proof is given in Appendix.)

The last lemma in particular yields the following property: there exist \(\epsilon _0 >0,\,\bar{h}>0\) and a constant \(\hat{m}>0\) independent of h such that

$$\begin{aligned} \hat{P}_{i}-\hat{R}_{i}\le -\hat{m},\quad \hat{P}_{i}+\hat{R}_{i}\ge \,\hat{m} \qquad \text{ for }\;\;t_{i}\in \,I_{\beta } \end{aligned}$$
(56)

whenever \(h<\bar{h}\) and  \(\Vert \hat{x}-x^{0}\Vert _{\infty }<\,\epsilon _0\) ; cf. (14), (15).

Lemma 11

Under assumptions (H0)–(H2), there exist indices \(\hat{k}, \hat{k}'\) such that, with the constant values \(u^1,u^2\) from Assumption (H1),

$$\begin{aligned} \hat{u}_{i}\equiv \,u^1,\;\;i=0,\dots ,\hat{k}, \qquad \hat{u}_{i}\equiv \,u^2,\;\;i=\hat{k}',\dots ,N-1, \end{aligned}$$

and   \(|t_{\hat{k}}-\tau _{s}|\,+\,|t_{\hat{k}'}-\tau '_{s}|=\mathrm{O}(h^{1/2})\).

Proof

Without loss of generality, consider the case \(u^1=-1\) only:

At the bang-singular junction point \(\tau _{s}\), both the function \(\sigma ^{0}\) and its time derivative vanish. Thus, for all j such that \(t_{j}<\tau _{s}\), it follows from (15) that

$$\begin{aligned} \sigma ^{0}(t_{j})=\int _{t_{j}}^{\tau _{s}}(\theta - t_{j})\ddot{\sigma }^{0}(\theta )\,d\theta \, \le \,-\frac{m^{2}}{2}(\tau _{s}-t_{j})^{2}. \end{aligned}$$

Consider \(\hat{\sigma }\):  if \(\hat{\sigma }_{j}< 0\) for all \(t_{j}<\tau _{s}\) then we can choose \(\hat{k}=k\) (see Lemma 2 and related notation). Otherwise, let \(\hat{k}\) be the first index such that \(\hat{\sigma }_{\hat{k}}\ge 0,\,t_{\hat{k}}<\tau _{s}\): from the previous lemma conclude

$$\begin{aligned} 0\,\le \, \hat{\sigma }_{\hat{k}}\le \,ch- \frac{m^{2}}{2}(\tau _{s}-t_{\hat{k}})^{2} \end{aligned}$$

and thus, the desired estimate for the left bang arc follows.

Analogously, the right bang arc is confirmed and estimated. \(\square \)

In the next step of analyzing the control structure it will be shown that, in a certain neighborhood of the state trajectory, the discrete controls \(\hat{u}\) consist of two bang arcs at right and left ends of the interval, and a singular arc located in the interior of \(I_{\beta }\). Following the ideas from [14], the proof will make use of auxiliary state-adjoint functions \(\hat{x}^{\pm }, \hat{p}^{\pm }\) which are obtained from solving (16), (17) with \(\delta =0\), constant \(u=u^{+}\equiv 1\) (or \(u=u^{-}\equiv -1\) respectively) and initial value \((\hat{x}_{i}, \hat{p}_{i})\) at given \(t=t_{i}\). Similarly, \(x^{\pm }, p^{\pm }\) solve (7), (8) with constant \(u\in \{-1,1\}\) and initial value \(({x}^{0}_{i}, {p}^{0}_{i})\) at \(t=t_{i}\). Notice that, for sufficiently small h, the assumptions on the input functions guarantee the existence of solutions at least for \(|t_{j}-t_{i}|\le \,\Delta _{0}\) (or \(|t-t_{i}|\le \,\Delta _{0}\) for the continuous version) for some \(\Delta _{0}>0\) independent of h.

Lemma 12

Let \(\bar{h}>0,\,\epsilon _0>0\) be the constants used for (56). Further, for given \(h<\bar{h}\), denote by \(\xi ^{h}_{0}=\hat{\xi }=(\hat{p},\hat{x},\hat{u},\hat{\mu })\) the solution of \((\mathrm{VI}^{h}_{0}\)) from Theorem 1. Then there exists a positive \(\epsilon _{1}<\epsilon _0\) with the following property: Whenever the state component \(\hat{x}\) satisfies  \(\Vert \hat{x}-x^{0}\Vert _{\infty }<\,\epsilon _{1}\), the control \(\hat{u}\) has bang-singular-bang structure. More precisely, there exist indices \(\hat{k}\) and \(\hat{k}'\) such that

$$\begin{aligned} \hat{u}_{i}=\left\{ \begin{array}{ccl} u^1 &{}\quad \text{ if } &{}\quad 0\le i\le \hat{k},\\ \hat{u}_{s,i}&{}\quad \text{ if } &{}\quad \hat{k}< i< \hat{k}',\\ {u}^2&{}\quad \text{ if } &{}\quad \hat{k}'\le i\le N-1. \end{array}\right. \end{aligned}$$
(57)

In addition,  \(|t_{\hat{k}}-\tau _{s}|+|t_{\hat{k}'}-\tau _{s}'|+\Vert \hat{u}_{s}-u^{0}_{s}\Vert _{(\infty , I_{\beta })}=\mathrm{O}(\epsilon _{1} + h)\).  The singular values are given by \(\hat{u}_{s,i}=-\hat{P}_{i}/\hat{R_{i}}\) and belong to \([m_{s}-1,1-m_{s}]\subset (-1,1)\) if \(\epsilon _{1}\) and h are taken sufficiently small. The constant \(m_{s}\) is independent of  h.

Proof

The proof is similar to that of [14, Theorem 3.1]; see Sect. 4 therein. For clarity reasons, the main arguments are repeated in the needed discrete formulation assuming (without loss of generality) that  \(u^1=-1\).

Lemma 11 says that \(\hat{u}\) has a bang arc in the left part of the interval [0, 1] where \(\hat{u}=-1\) and thus,  \(\hat{\sigma }=\hat{\sigma }^{-}\) (continued from \(t_{0}=0\), e.g.) as long as \(\hat{\sigma }^{-}<0\). If h is sufficiently small, the right end of the arc lies inside \(I_{\beta }\).

Now, from Lemma 10 we have

$$\begin{aligned} \Vert \Delta ^{1}(\hat{\sigma }^{-}-\sigma ^{-})\Vert _{\infty }= \text{ O }(\epsilon _{1} +h). \end{aligned}$$

and \(\Delta ^{1}\hat{\sigma }^{-}\) (as well as the finite difference approximation of the time derivative of \({\sigma }^{-}\)) changes its sign, i.e., \((\Delta ^{1}\hat{\sigma }^{-})_{i}\le 0\) for \(i\ge {j}\) and some node \(t_{{j}}\) near \(\tau _{s}\): Due to (15) and (56), \(\Delta ^{2}\hat{\sigma }^{-}\le -\hat{m}<0\) so that  \(|t_{{j}}-\tau _{s}|=\)O\((\epsilon _{1} +h)\) follows.

Let j be the first index where \((\Delta ^{1}\hat{\sigma }^{-})_{i}\le 0\). We will distinguish between the following cases:

Case 1 \(\hat{\sigma }^{-}_{{j}}<\,0\).

In this case, \(\hat{u}_{j}=-1,\;(\Delta ^{2}\hat{\sigma })_{j}\le \,-\hat{m}<0\) and \(\hat{\sigma }^{-}\) is concavely decreasing for \(t_{i}>t_{j}\). Therefore,  \(\hat{\sigma }_{i}=\hat{\sigma }^{-}_{i}\) remains to be valid at least for all i with \(t_{i}-t_{j}\le \Delta _{0}\). We obtain:

$$\begin{aligned} (\Delta ^{1}\hat{\sigma })_{j+\nu }\le & {} -\hat{m}\cdot \nu \,h,\\ \hat{\sigma }_{j+\nu }< & {} h\sum _{k=1}^{\nu -1}(\Delta ^{1}\hat{\sigma })_{j+k} \le \,-\frac{\hat{m}}{2}(\nu -1)^{2}h^{2}. \end{aligned}$$

If \(t_{i}\) is the last node on the interval \([t_{j}, t_{j}+\Delta _{0}]\) then \(\hat{\sigma }_{i}\le \,-\hat{m}\,(\Delta _{0}-2h)^{2}/2\) contradicting the conditions \(\sigma ^{0}(t)=0\) for \(t\ge \tau _{s}, |t_{j}-\tau _{s}|=\hbox {O}(\epsilon _{1}+ h)\),   and \(\Vert \hat{\sigma }-\sigma _{0}\Vert _{\infty }=\hbox {O}(h)\)  from Lemma 10.

Case 2   \(\hat{\sigma }^{-}_{{j}}>\,0\).

In this case, \(\hat{\sigma }\ge 0\) has appeared first for some node \(t_{j'}\le t_{j}\): if \(\hat{\sigma }_{{j'}}>\,0\), the function coincides with the convexly increasing function \(\hat{\sigma }^{+}\) for \(t_{j'}\le t_{i}\le t_{j'}+\Delta _{0}\). Estimating \(\hat{\sigma }^{+}_{j'+\nu }\) in analogy to Case 1, we end up with a contradiction. If \(\hat{\sigma }^{-}_{{j'}}=\,0\) then proceed as in Case 3 below.

Case 3   \(\hat{\sigma }^{-}_{{j}}=\,0\).

It is sufficient to consider \((\Delta ^{1}\hat{\sigma })_{j}=0\) since \((\Delta ^{1}\hat{\sigma })_{j}\not =0\) yields \(\hat{\sigma }_{{j+1}}\not =\,0\) and thus, a situation as in Case 1 or Case 2. (Similarly, one can exclude \((\Delta ^{1}\hat{\sigma })_{j+1}\not =0\) by contradiction).

For \((\Delta ^{1}\hat{\sigma })_{j}=(\Delta ^{1}\hat{\sigma })_{j+1}=0\), obviously \((\Delta ^{2}\hat{\sigma })_{j+1}=0\) so that \(\hat{u}_{j+1}=\hat{u}_{s,j+1}\). Whenever \(t_{i}\le 0.5(\tau _{s}+\tau _{s}')+\bar{h}\), a change to positive resp. negative values of \(\hat{\sigma }\) at \(t_{i}\) can be excluded by repeating the arguments above. Thus, the left bang arc is concatenated to a central singular arc.

The analysis can be carried out similarly from the right end, and the statement on the control structure together with

$$\begin{aligned} |t_{{j}}-\tau _{s}|\,+\, |t_{{j}'}-\tau '_{s}|=\,\text{ O }(\epsilon _{1} +h) \end{aligned}$$

follows. Finally, by use of Lemma 10, the estimate for \(\hat{u}_{s}\) is obtained. \(\square \)

The structural information allows to strengthen the general result from Theorem 1 and formulate a convergence result for the solutions of (VI\(^{h}_{0}\)) w.r.t. (discrete) \(L_{\infty }\times L_{\infty }\times L_{1}\times L_{\infty }\) topology:

Theorem 2

Let the assumptions (H0)–(H2) hold for the solution \(\xi ^{0}=(p^{0},x^{0},u^{0},\mu ^{0})\) of (VI). Further, suppose \({h}', \epsilon , \bar{\rho }'\) and the set \(V_\epsilon \) be given as in Theorem 1. If \(h<{h}'\) is small enough to ensure \(\Vert \tilde{\delta }^{h}\Vert _{D}<\,\bar{\rho }'\) then the following statements are true:

  1. (i)

    On \(V_{\epsilon }\), the discrete problem (\(\hbox {VI}^{h}_{0}\)) has a unique solution \(\xi ^{h}_{0}=\hat{\xi }=(\hat{p},\hat{x},\hat{u},\hat{\mu })\) estimated by

    $$\begin{aligned} \Vert \hat{x}-x^{0}\Vert _{2}+\,\Vert \hat{p}-p^{0}\Vert _{\infty }+\,\Vert \hat{\mu }-\mu ^{0}\Vert _{\infty } =\,\text{ O }(h), \end{aligned}$$

    and   \(\Vert \hat{x}-x^{0}\Vert _{\infty }=\)O\((h^{1/2})\).

  2. (ii)

    There exists a constant \(h_{1}<{h}'\) such that, for each \(h<h_{1}\), the control \(\hat{u}\) has bang-singular-bang structure in the sense of (57). The state-control pair and associated bang-singular junction points \(\hat{\tau },\;\hat{\tau }'\) satisfy

    $$\begin{aligned} \Vert \hat{x}-x^{0}\Vert _{\infty }+\,\Vert \hat{u}-u^{0}\Vert _{1}+ \,|\hat{\tau }-\tau _{s}|\,+\,|\hat{\tau }'-\tau _{s}'| \,=\,\text{ O }(h). \end{aligned}$$

Proof

The first estimates from part (i) are a direct consequence of Theorem 1 applied to the choice \(\delta =0\), and of Lemma 10 (see also the proof of the latter in Appendix).  In order to obtain the estimate for \(\Vert \hat{x}-x^{0}\Vert _{\infty }\) we observe that, under assumption (H0) on the data, there exists a constant \(M_{1}\) independent of h such that

$$\begin{aligned} \Vert \hat{x}-x^0\Vert _\infty +\,\Vert \Delta ^1(\hat{x}-x^0)\Vert _2\le \,M_1. \end{aligned}$$

Applying Lemma 13 from Appendix, the desired estimate follows.

For proving part (ii), the technique from [15, Lemma 4.1] will be adapted. To this aim, define

$$\begin{aligned} y^{0}_{i}=\,h\sum _{j=0}^{i-1}u^{0}(t_{j}+0),\qquad \hat{y}_{i}=\,h\sum _{j=0}^{i-1}\hat{u}_{j}. \end{aligned}$$

Notice that \(y^{0}\) slightly differs from \(\tilde{y}^{h}\) defined via Goh transformation with the input \(\tilde{u}^{h}\), (21). Assumptions (H1) and (H2), however, guarantee that the pointwise differences do not exceed O(h). By Theorem 1 and Lemma 2, we therefore have

$$\begin{aligned} \Vert \hat{y}-y^{0}\Vert _{1}\le \,\Vert \hat{y}-\tilde{y}^{h}\Vert _{2}+\, \Vert \tilde{y}^{h}-y^{0}\Vert _{\infty }=\; \text{ O }(h). \end{aligned}$$
(58)

Similarly obtain  \(|\hat{y}_{N}-y^{0}_{N}|=\hbox {O}(h)\).

In part (i) it was particularly proved that \(\Vert \hat{x}-x^{0}\Vert _{\infty }\rightarrow \;0\) as \(h\rightarrow 0\). Hence, there exists a positive \(h_{1}<\bar{h}\) such that \(\Vert \hat{x}-x^{0}\Vert _{\infty }<\,\epsilon _{1}\) holds with the constant \(\epsilon _{1}\) from Lemma 12 for \(h<h_{1}\), and \(\hat{u}\) has the structure (57). Therefore, the y–differences given at nodes \(t_{i}\) by

$$\begin{aligned} \hat{y}_{i}-y^{0}_{i}=h\sum _{j=0}^{i-1}(\hat{u}_{j}-u^{0}_{j}) \,=\, (\hat{y}_{N}-y^{0}_{N})\,-\,h\sum _{j=i}^{N-1} (\hat{u}_{j}-u^{0}_{j}) \end{aligned}$$

can be estimated, in particular, for points \(t_{i}\) from the intersection \((\tau _{s},\tau _{s}')\cap (\hat{\tau },\hat{\tau }')\) as follows: in case \(\tau _{s}<\hat{\tau }\) the above formula yields

$$\begin{aligned} | \hat{y}_{i}-y^{0}_{i}|= & {} h\,\left| \sum _{j:\,\tau _{s}<t_{j}<\hat{\tau }}(u^1-u^{0}_{s,j})\,+\, \sum _{j: \hat{\tau }\le t_{j}<t_{i}}(\hat{u}_{s,j}-u^{0}_{s,j})\right| \nonumber \\\ge & {} \min \{m,\hat{m}\}\,|\hat{\tau }-\tau _{s}|\, -\,\Vert \hat{u}_{s}-u^{0}_{s}\Vert _{(1,I_{\beta })} \end{aligned}$$
(59)

(and the same relation results in case \(\tau _{s}\ge \hat{\tau }\), too). Analogously obtain

$$\begin{aligned} | \hat{y}_{i}-y^{0}_{i}|\;\ge \; \min \{m,\hat{m}\} \,|\hat{\tau }'-\tau '_{s}|\,-\,\Vert \hat{u}_{s}-u^{0}_{s}\Vert _{(1,I_{\beta })} -|\hat{y}_{N}-y^{0}_{N}|. \end{aligned}$$
(60)

For \(h<h_{1}\), the singular control \(\hat{u}_{s}\) restricted to nodes \(t_{i}\in I_{\beta }\) is well defined. Further, from (14), (56) obtain \(R_{i}\ge m>0,\;\hat{R}_{i}\ge \hat{m}>0,\;t_{i}\in I_{\beta },\) so that

$$\begin{aligned} \Vert \hat{u}_{s}-u^{0}_{s}\Vert _{(1,I_{\beta })}\le \,\hat{m}^{-1}\Vert \hat{P}-P\Vert _{1}+(m\hat{m})^{-1}\Vert P\Vert _{\infty }\Vert \hat{R}-R\Vert _{1}=\;\text{ O }(h) \end{aligned}$$

follows from (H0) and Lemma 10.

Combining the estimates (59), (60), we come to the following conclusion: there exists a constant \(c>0\) independent of h such that

$$\begin{aligned} |\hat{\tau }-\tau _{s}|\,+\,|\hat{\tau }'-\tau '_{s}|\le c\,\left( \Vert \hat{y}-y^{0}\Vert _{1}+\Vert \hat{u}_{s}-u^{0}_{s}\Vert _{(1,I_{\beta })} +\,|\hat{y}_{N}-y^{0}_{N}|\right) = \text{ O }(h). \end{aligned}$$

Finally, by \(0\le \,\hat{u}_{i},u^{0}_{i}\le 1\) for \(i=0,\dots ,N-1\),

$$\begin{aligned} \Vert \hat{u}-u^{0}\Vert _{1}\le \, \Vert \hat{u}_{s}-u^{0}_{s}\Vert _{(1,I_{\beta })}+ |\hat{\tau }-\tau _{s}| +|\hat{\tau }'-\tau '_{s}| =\text{ O }(h) \end{aligned}$$

follows. Remembering (16), for \(\delta =0\) this last estimate also yields \(\Vert \hat{x}-x^{0}\Vert _{\infty }=\)O(h)  and hence, the theorem. \(\square \)

6 Numerical test

The discretization approach will be illustrated by approximating the solution of the following problem,

figure f

As it was shown in [14], for small \(\delta \) the solution is of bang-singular-bang structure, and the control vanishes along the singular arc. For data \(\delta =0.15, a=3\), the optimal termination time for (61) is found from a synthesis approach as \(T^{*}=3.5117\). The terminal time T for (P\(_{\delta }\)) is chosen as \(T=3.2\,<\,T^{*}\). Assuming the same principal control structure as in time-optimal termination case, one can find piecewise analytic expressions for x, p and determine the bang-singular junction points  \(\tau _{s}=2.1482, \;\tau '_{s}=2.6932\) for (\(\hbox {P}_{\delta }\)): this defines the reference solution \(u^{0}\).

In the numerical tests, we replace (\(\hbox {VI}^{h}_{0}\)) by an equivalent system of equations: for \(i=1,\dots ,N\), the complementarity relations (18), (19) are expressed by

$$\begin{aligned} \phi _{ \gamma }(\mu _{1i},1-u_{i})=0,\quad \phi _{\gamma }(\mu _{2i},1+u_{i})=0, \end{aligned}$$

where   \(\phi _{\gamma }(w,\nu )=w+\nu \,-\sqrt{w^{2}+\,\nu ^{2}+\gamma }\) with \(\gamma =0\) stands for the Fischer–Burmeister function [16, 17]. Notice that the original function \(\phi _{0}\) is non-differentiable at the origin. In order to ensure robustness of the iterative solution when smooth solvers like the matlab Footnote 1 routine fsolve are used, the additional regularization parameter \(\gamma >0\) was introduced (cf. [25] and the survey [18]). Further, the system (\(\hbox {VI}^{h}_{0}\)) was reduced to a problem of finding \((u,\mu _{1},\mu _{2})\). To this aim, the system (16), (17) is solved explicitely and the resulting vector \(p=p(u)\) is inserted into (18). The final problem consists in solving \(\bar{F}^{\gamma }(u,\mu _{1},\mu _{2})=0\) where

$$\begin{aligned} \bar{F}^{\gamma }_{1i}=\,B^{T}p_{i}(u)- \mu _{1i}+\mu _{2i},\quad \bar{F}^{\gamma }_{2i}=\,\phi _{\gamma }(\mu _{1i},1-u_{i}),\quad \bar{F}^{\gamma }_{3i}=\,\phi _{\gamma }(\mu _{2i},1+u_{i}). \end{aligned}$$

For the initial values used in fsolve, for given \(\bar{u}\) find \((\bar{x},\bar{p})=(x(\bar{u}), p(\bar{u}))\) from the canonical system (16), (17), and \(\bar{\mu }_{1,2}=\mu _{1,2}(\bar{u})\) as positive resp. negative parts of \(\sigma =B^{T}p(\bar{u})\). Afterwards, the vector \((\bar{u},\bar{\mu }_{1},\bar{\mu }_{2})\) is projected onto the closed set \(\{[-1+\gamma ,1-\gamma ]^{N}\times [\gamma ,\infty )^{2N}\}\): this change turns \(\bar{u}\) and \(\bar{\mu }_{1,2}\) into strictly inner points of their respective domain of definition and again, improves the robustness of the solution process.

Table 1 Approximation errors for control function u and junction times \(\hat{\tau },\,\hat{\tau }'\)
Full size table

Table 1 summarizes selected results. The initial control data are \(\bar{u}\equiv 1\). (It should be mentioned that tests with \(\bar{u}\equiv 0.5\) as well as for some prescribed bang-singular-bang regimes, showed essentially the same behavior.) One can see that the solution process gives fairly good approximations of the reference control. Let us mention that, in all reported cases, the discrete controls have clearly visible bang arcs at left and right ends of the time interval and an inner singular arc with \(u=0\) (up to marginal rounding errors perhaps). Due to the regularization by \(\gamma >0\), but also by rounding effects, the bang-singular junction appears as an intermediate monotonicity interval rather then a “sharp” jump but is restricted to few nodes: with \(m_{\epsilon }=0.1 \,\Vert u-u^{0}\Vert _{1}\), we determined two intervals \(I_{s}, I'_{s}\) where   \(1-m_{\epsilon }\ge \,u_{i}\ge \,m_{\epsilon }\). The approximation error \(\Delta \tau \) for \(\tau _{s}, \tau '_{s}\) is defined as the maximum of distances from \(\tau _{s}\) to interval ends of \(I_{s}\), and from \(\tau '_{s}\) to interval ends of \(I'_{s}\).

Finally notice that, in the given example, the solution could be also found for \(\gamma =0\): in all tests the junction points are approximated with accuracy \(\Delta \tau < 1.5\,h\), i.e., the “1–0–1” switching structure of the discrete control becomes evident. For \(N=200\), the control is approximated with a remaining tolerance bound of \(\Vert u-u^{0}\Vert _{1}=\,0.0012\).