This paper addresses control problems for dynamic systems with a vector-valued criterion. A vector Hamiltonian formalism for solving optimal control problems with several independent value functionals was described in [1, 2]. Below, we construct estimates for systems with a dynamic equation and a vector-valued criterion that involves a bounded disturbance.

The study of such problems requires the construction of guaranteed minimax estimates. For this purpose, we introduce the concepts of set-valued minimax and maximin for a vector-valued criterion and investigate the relations between them. It is shown below that an inequality between the minimax and maximin in the one-dimensional case may be violated for higher dimensional criteria. Moreover, this effect is regular and may be observed even for the simplest functionals.

In Section 1, we introduce the basic definitions, including the concepts of set-valued minimum, maximum, minimax, and maximin. Functionals with separated variables, i.e., those representable in the form of sums of two vector-valued functionals that possibly depend on different vector-valued variables are studied in Section 2. Bilinear vector-valued functionals in the general form are considered in Section 3.

1 DEFINITIONS

Let us introduce the basic concepts used in what follows. As an order relation for comparing vectors of the space \({{\mathbb{R}}^{p}}\), we use the Pareto order.

Definition 1. We say that a vector \(z \in {{\mathbb{R}}^{p}}\)is dominated by a vector \(\hat {z} \in {{\mathbb{R}}^{p}}\) in the sense of Pareto if they are different and their components are related by the inequalities

$${{\hat {z}}_{i}} \leqslant {{z}_{i}},\quad i = 1,\; \ldots ,\;p.$$

This relation is denoted as \(\hat {z} \leqslant z\).

Definition 2. Consider a mapping \({\mathbf{Z}}:X \to {{\mathbb{R}}^{p}}\). The set-valued minimum of the set of its values is defined as the union of all Pareto nondominated points of its image:

$${\mathbf{Min}}\;{\mathbf{Z}}(X) = \{ {{z}_{*}} \in {\mathbf{Z}}(X)\left| {{\text{ }}z \in {\mathbf{Z}}(X),z \ne {{z}_{*}}:\;{{z}_{*}}} \right. \leqslant z\} .$$

The set-valued maximum of Z(X) is the set of all Pareto non-dominating values:

$${\mathbf{Max}}\;{\mathbf{Z}}(X) = \{ z\text{*} \in {\mathbf{Z}}(X)\left| {{\text{ }}z \in {\mathbf{Z}}(X),z \ne z\text{*}:\;z\text{*} \geqslant z} \right.\} .$$

Now consider a mapping \({\mathbf{F}}{\text{:}}\;U \times V \to {{\mathbb{R}}^{p}}\). Assume that all vector-valued functionals considered below satisfy the following regularity conditions:

(i) \({\mathbf{F}}(U,{v})\) are closed for all \({v} \in V\);

(ii) \({\mathbf{F}}(u,V)\) are closed for all \(u \in U\);

(iii) there exist \({{M}_{*}},M\text{*} \in {{\mathbb{R}}^{p}}\) such that \({{M}_{*}}\) ≤ \({\mathbf{F}}(u,{v})\) ≤ M* simultaneously for all (u, \({v})\) ∈ U × V.

These conditions guarantee the existence of a Pareto frontier for all images considered in what follows. Note that no additional assumptions are made about the domain of \({\mathbf{F}}(u,{v})\), since the obtained results are valid for both finite- and infinite-dimensional U and V.

Definition 3. The set-valued minimax for the values of \({\mathbf{F}}(u,{v})\) on the set U × V is defined as

$${\mathbf{Min}_{u}}{\mathbf{Max}_{{v}}}{\mathbf{F}}(u,{v}) = {\mathbf{Min}}\left( {\bigcup\limits_u {{\mathbf{MaxF}}(u,V)} } \right).$$

The set-valued maximin for \({\mathbf{F}}(u,{v})\) on U × V is defined as

$${\mathbf{Ma}}{{{\mathbf{x}}}_{{v}}}{\mathbf{Mi}}{{{\mathbf{n}}}_{u}}{\mathbf{F}}(u,{v}) = {\mathbf{Max}}\left\{ {\bigcup\limits_{v} {{\mathbf{Min}}} \,{\mathbf{F}}(U,{v})} \right\}.$$

Thus, in the general case, the set-valued minimax and maximin are both represented by sets. Since they are supposed to be compared in what follows, we define how to understand inequality between two sets.

Definition 4. Two sets \(A,B \subset {{\mathbb{R}}^{p}}\) are said to satisfy the inequality \(A \leqslant B\) if

$$\forall b \in B{\backslash }A \Rightarrow \exists a \in A:a \leqslant b.$$

Definition 5. With the use of the introduced notation, the relation

$${\mathbf{Ma}}{{{\mathbf{x}}}_{{v}}}{\mathbf{Mi}}{{{\mathbf{n}}}_{u}}{\mathbf{F}}(u,{v}) \leqslant {\mathbf{Mi}}{{{\mathbf{n}}}_{u}}{\mathbf{Ma}}{{{\mathbf{x}}}_{{v}}}{\mathbf{F}}(u,{v})$$
(1)

is called the basic minimax-maximin inequality for the Pareto order, while the reverse minimax-maximin inequality is understood as

$${\mathbf{Mi}}{{{\mathbf{n}}}_{u}}{\mathbf{Ma}}{{{\mathbf{x}}}_{{v}}}{\mathbf{F}}(u,{v}) \leqslant {\mathbf{Ma}}{{{\mathbf{x}}}_{{v}}}{\mathbf{Mi}}{{{\mathbf{n}}}_{u}}{\mathbf{F}}(u,{v}).$$
(2)

As the dimensionality of functional values is increased, the relations between minimax and maximin change qualitatively and, in contrast to the one-dimensional case, the reverse minimax-maximin inequality becomes not only reachable, but also, generally speaking, regular. Below, we consider two different general vector-valued functionals and give necessary conditions for inequalities (1) and (2) to hold for each of them.

2 FUNCTIONALS WITH SEPARATED VARIABLES

Consider a functional of the form \({\mathbf{S}}(u,{v})\) = Φ(u) + Ψ(\({v}\)). For such functionals, we show that either the set-valued minimax and maximin are equal to each other or the reverse minimax-maximin inequality (2) is true. Indeed, using the definition and Lemma 2 from [2], we obtain

$$\begin{gathered} {\mathbf{Mi}}{{{\mathbf{n}}}_{u}}{\mathbf{Ma}}{{{\mathbf{x}}}_{{v}}}{\mathbf{S}}(u,{v}) \\ = {\mathbf{Min}}\left\{ {{\mathbf{\Phi }}(\tilde {u}) + {\mathbf{Max}}\;{\mathbf{\Psi }}(V)\left| {\tilde {u} \in U} \right.} \right\} \\ = \;{\mathbf{Min}}\left\{ {{\mathbf{Min}}\;{\mathbf{\Phi }}(U) + {\mathbf{Max}}\;{\mathbf{\Psi }}(V)} \right\}. \\ \end{gathered} $$

Similarly,

$$\begin{gathered} {\mathbf{Ma}}{{{\mathbf{x}}}_{{v}}}{\mathbf{Mi}}{{{\mathbf{n}}}_{u}}{\mathbf{S}}(u,{v}) \\ = {\mathbf{Max}}\left\{ {{\mathbf{Min}}\;{\mathbf{\Phi }}(U) + {\mathbf{Max}}\;{\mathbf{\Psi }}(V)} \right\}. \\ \end{gathered} $$

Thus, assuming that a set-valued minimax and a set-valued maximin for \({\mathbf{F}}(u,{v})\) exist, we conclude that the reverse minimax-maximin inequality (2) is always true for functionals of the indicated form.

Moreover, the inequality holds as an equality where the set

$${\mathbf{Min}}\;{\mathbf{\Phi }}(U) + {\mathbf{Max}}\;{\mathbf{\Psi }}(V)$$

coincides with its set-valued maximum and minimum. This property holds if at least one of the sets \({\mathbf{Min}}\;{\mathbf{\Phi }}(U)\) and \({\mathbf{Max}}\;{\mathbf{\Psi }}(V)\) is a singleton. Additionally, if both terms lie in the same hyperplane, we also have the equality.

Thus, the reverse minimax-maximin inequality (2) holds for a sufficiently large class of vector-valued functionals; the basic inequality (1) holds for them only if the corresponding set-valued minimax and maximin are equal to each other.

3 BILINEAR VECTOR-VALUED FUNCTIONALS

Consider a functional of the form

$${\mathbf{B}}(u,{v}) = \left[ {\begin{array}{*{20}{c}} {\left\langle {u,{{A}_{1}}{v}} \right\rangle } \\ \vdots \\ {\left\langle {u,{{A}_{p}}{v}} \right\rangle } \end{array}} \right],$$

where 〈x, y〉 denotes the scalar product of elements x and y and Ai are real matrices. It is assumed that \(\left\langle {u,{{A}_{i}}{v}} \right\rangle \) are defined for all \(u \in U\), \({v} \in V\), and i = 1, ..., p.

If \(V \subset \mathbb{R}\) (or \(U \subset \mathbb{R}\)), then the equality between the minimax and maximin always holds for vector-valued functionals of the indicated form. If none of the components is one-dimensional, an additional study is required.

To determine the conditions under which the basic minimax-maximin inequality holds for the bilinear vector-valued functional \({\mathbf{B}}(u,{v})\), we formulate the conditions for its violation and require that they never hold.

Proposition 1. Given a vector-valued functional \({\mathbf{F}}(u,{v}){\text{:}}\;U \times V \to {{\mathbb{R}}^{p}}\)satisfying the above-described regularity conditions, suppose that there exist points \(f\text{*} = {\mathbf{F}}(u\text{*},{v}\text{*})\) ∈ \({\mathbf{Mi}}{{{\mathbf{n}}}_{u}}{\mathbf{Ma}}{{{\mathbf{x}}}_{{v}}}{\mathbf{F}}(u,{v})\)and \({{f}_{*}}\) = \({\mathbf{F}}({{u}_{*}},{{{v}}_{*}}) \in {\mathbf{Ma}}{{{\mathbf{x}}}_{{v}}}{\mathbf{Mi}}{{{\mathbf{n}}}_{u}}{\mathbf{F}}(u,{v})\)for which the basic minimax-maximin inequality (1) is violated:

$$f\text{*} \leqslant {{f}_{*}}.$$

Then the point \(\hat {f} = {\mathbf{F}}(u\text{*},{{{v}}_{*}})\)is comparable (in the sense of Pareto) with neither \(f{\text{*}}\)nor \({{f}_{*}}\).

Remark 1. The last proposition holds not only for the Pareto order. If the Pareto order is considered in the space \({{\mathbb{R}}^{p}}\) of functional values, then the condition for the violation of the basic minimax-maximin inequality may be obtained in a more explicit form.

Corollary 1. Suppose that the conditions of Proposition 1 are satisfied. Then there exist i ≠ j and \(k \ne l,\)where \(i,j,k,l = 1,\; \ldots ,\;p\)such that

$$\begin{array}{*{20}{c}} {({{F}_{i}}(u\text{*},{v}\text{*}) - {{F}_{i}}(u\text{*},{{{v}}_{*}}))({{F}_{j}}(u\text{*},{v}\text{*}) - {{F}_{j}}(u\text{*},{{{v}}_{*}})) < 0,} \\ {({{F}_{k}}({{u}_{*}},{{{v}}_{*}}) - {{F}_{k}}(u\text{*},{{{v}}_{*}}))({{F}_{l}}({{u}_{*}},{{{v}}_{*}}) - {{F}_{l}}(u\text{*},{{{v}}_{*}})) < 0.} \end{array}$$

Using this result and the Sylvester criterion for semidefinite matrices (for more details, see [4]), we can establish the following sufficient condition for the basic minimax-maximin inequality to be true for an arbitrary bilinear functional \({\mathbf{B}}(u,{v})\).

Proposition 2. Suppose that, for all \(i,j = 1,\; \ldots ,\;p\), all corner minors of order k ≤ 2 of the matrices

$${{B}_{{ij}}} = {{A}_{i}}{vv}'A_{j}^{'} + {{A}_{j}}{vv}'A_{i}^{'}$$

are positive semidefinite simultaneously for all \({v} \in V - V\). Then the basic minimax-maximin inequality (1) is true for the bilinear functional \({\mathbf{B}}(u,{v})\).

Writing the diagonal elements of the matrices Bij in explicit form, we can obtain the following assertion, which allows one to draw conclusions about whether the conditions of the proposition are satisfied.

Corollary 2. Suppose that the conditions of Proposition 2 are satisfied. Then, for all \(i,j = 1,\; \ldots ,\;p\), the components of the matrices Ai and Aj satisfy the condition

$${{[{{A}_{i}}]}_{{kl}}}{{[{{A}_{j}}]}_{{kl}}} \geqslant 0,$$

where \({{[{{A}_{i}}]}_{{kl}}}\)denotes the element of Ai in the kth row and the lth column.

Additionally, by using Corollary 1, we can obtain a sufficient condition for the violation of the basic minimax-maximin inequality in the special case of a vector functional consisting of only two criteria.

Corollary 3. Suppose that \({\mathbf{B}}(u,{v}) \in {{\mathbb{R}}^{2}}\)and, for all \(u \in U - U\)and \({v} \in V - V\),

$$\begin{array}{*{20}{c}} {{{A}_{1}}uu'A_{2}^{'} < 0,} \\ {{{A}_{1}}{vv}'A_{2}^{'} < 0.} \end{array}$$

Then the reverse minimax-maximin inequality (2) is true for the vector-valued functional \({\mathbf{B}}(u,{v})\).

Thus, for the case p = 2, we have obtained a necessary and sufficient condition for the violation of the basic minimax-maximin inequality.

CONCLUSIONS

The concepts of set-valued minimax and maximin for vector-valued functionals over the real field were introduced. The basic and reverse minimax-maximin inequalities relating these concepts were formulated.

For vector-valued functionals with separated variables, it was shown that the reverse minimax-maximin inequality may be true for whole classes of value functionals.

A necessary condition for the violation of the basic minimax-maximin inequality was obtained. Additionally, necessary and sufficient conditions for it to be true in the case of bilinear vector-valued functionals were examined.