Abstract
If is a finite-dimensional Hilbert space, then Lyapunov proved that the mild solution {u x(t)} of the Cauchy problem \(\dot{u} (t) = Au(t)\), with the initial condition satisfies the following condition (a) , with \(r,\;c_{0}>0\), for t>0, if and only if, there exists a positive definite matrix R satisfying two conditions: (i) , , \(c_{1}\; c_{2}>0\), and (ii) A ∗ R+RA=−I. The infinite dimensional analogue of this theorem is given by Datko. However, in this case the operator R does not satisfy the lower bound in condition (i). Lyapunov uses this lower bound crucially in going from linear to non-linear case. The solution in non-linear case will satisfy condition (a), termed exponential stability. When the operator A generates a pseudo-contraction semigroup, we produce an operator R satisfying both conditions (i) and (ii) in a deterministic equation. At this stage, we discuss stochastic equations and demonstrate how to study exponential stability for mild solutions of non-linear equations. We cannot use the Itô formula for mild solutions of stochastic differential equations, thus we have to approximate them using Yosida approximation, which produces a sequence of strong solutions. We use this approximation to obtain a sufficient condition for exponential stability in the mean square sense (m.s.s.) when the Lyapunov function Ψ exists. However, in order to use this approximation, we need to assume that Ψ is locally bounded. In the linear case we produce such a Lyapunov function obtaining necessary and sufficient conditions for the exponential stability in the m.s.s. of the solution. Then the first order approximation is used to study the exponential stability in the m.s.s. of mild solutions to non-linear equations following the ideas of Lyapunov. We provide the stochastic analogue of Datko’s theorem in the Appendix.
We also present Lyapunov function approach for studying exponential stability in the m.s.s. of strong variational solutions in a Gelfand triplet of real separable Hilbert spaces V↪H↪V ∗. In the deterministic linear case, A:V→V ∗ is a bounded operator satisfying the coercivity and monotonicity conditions. Then the solution of the Cauchy problem u x(t) is in C([0,T],H) We prove that u x(t) is exponentially stable if and only if there exists an operator \({\tilde{C}}\) satisfying conditions (i) and (ii). However the norm in V is used to define \({\tilde{C}}\) through the equality \(\langle {\tilde{C}}x,x\rangle_{H} = \int_{0}^{\infty}\| u^{x}(t)\|^{2}_{V}\, dt\). In order to show that \(\langle {\tilde{C}}x,x\rangle_{H}\) is finite we need the coercivity condition. We follow the same plan as for mild solutions for stochastic equations. We obtain a sufficient condition for exponential stability in the m.s.s. in the non-linear case in terms of the existence of the Lyapunov function. For linear equations we obtain necessary and sufficient conditions for exponential stability in the m.s.s. by constructing the Lyapunov function. We extend that result to non-linear equations using the first order approximation of non-linearities.
We introduce the concept of stability in probability for solutions to stochastic equations and provide a sufficient condition in terms of the existence of the Lyapunov function. Using the previous results, we show that exponential stability in the m.s.s. implies stability in probability.
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© 2011 Springer-Verlag Berlin Heidelberg
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Gawarecki, L., Mandrekar, V. (2011). Stability Theory for Strong and Mild Solutions. In: Stochastic Differential Equations in Infinite Dimensions. Probability and Its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16194-0_6
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DOI: https://doi.org/10.1007/978-3-642-16194-0_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-16193-3
Online ISBN: 978-3-642-16194-0
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