Evgenios P. Avgerinos

Random nonlinear evolution inclusions in reflexive Banach spaces

In this paper we present two existence results for a large class of random, nonlinear, multivalued evolution equations defined in a reflexive, separable Banach space and involving an m-dissipative operator. Applications to random multivalued parabolic p.d.e.s are presented. Our work unifies and extends earlier results of Kampe de Feriet, Gopalsamy and Bharucha-Reid, Becus and Itoh. 1. Introduction. Many problems in physics, engineering, biology and social sciences lead to mathematical equations. In these equations, the coefficients and the other parameters have their origin in experimental data and represent some kind of average value. Therefore in many instances due to wide variations of the data or even due to our own ignorance, it is appropriate to abandon the deterministic model in favor of a stochastic one. This is very nicely exemplified in the books of Bharucha-Reid (4) and Soong (21). In this paper we present two existence results for a class of random nonlinear multivalued evolution equations defined in a reflexive, separable Banach space. This class of evolution equations models linear and several nonlinear partial differential equations of parabolic type. So our work unifies and extends earlier works on random parabolic partial differential equations. In particular, Kampe de Feriet (12, 13) was the first to study the random heat equation. The randomness entered in the problem through the initial value data. Later Gopalsamy and Bharucha-Reid (10) and Becus (2) introduced also randomness in the boundary value and source terms. Finally Itoh (11), allowed randomness to appear also in the operator, which instead of the Laplacian, was a general random, single valued, everywhere defined continuous, accretive operator. Our results cover all the above-mentioned works. Let (I2, E) be a measurable space and X a separable Banach space. Through- out this work we will be using the following notations: P^C)(X) = {A C X: nonempty, closed, (convex)} and Pkc(X) = {A C X: nonempty, compact, convex}. A multifunction F: fi —► Pf(X) is said to be measurable if for every x G X,

Variational stability of infinite dimensional optimal control problems

We examine the variational stability of infinite dimensional optimal control problems governed by non-linear evolution equations. Our tools are the Kuratowski-Mosco convergence of sets and the corresponding τ-convergence of functions. We prove the τ-convergence of cost functionals, the convergence of the values of the problems and we examine the variational stability of the solution and reachable sets. These results are then applied to a sequence of non-linear parabolic distributed parameter optimal control problems.

Solutions and Periodic Solutions for Nonlinear Evolution Equations with Non-Monotone Perturbations

In this paper we solve periodic and Cauchy problems for nonlinear evolution equations driven by time-dependent, pseudomonotone operators and a non-monotone perturbation term. Our proof produces as a by-product a useful property of the solution map for maximal monotone problems. Two examples of nonlinear parabolic problems illustrate the applicability of our work.

On the sensitivity and relaxability of optimal control problems governed by nonlinear evolution equations with state constraints

In this work we examine the relation between the sensitivity (well posedness) and relaxability for nonlinear distributed parameter systems. We introduce the notion of “strong calmness” which describes the dependence of the value of the problem on the state constraints, and we show that it is equivalent to “relaxability”. We also present an alternative control free description of the relaxed problem. An example of a nonlinear parabolic optimal control problem of Lagrange type is worked out in detail.

On U(·)-invariance for finite and infinite dimensional control systems

The purpose of this paper is to study the property of set invariance in connection with finite and infinite dimensional, nonlinear control systems. Our work was motivated by the papers of Feuer-Heymann [5], [6], who investigated this problem in the context of nonlinear, finite dimensional control systems. Using some recent results and techniques from the theory of multifunctions and the theory of differential inclusions, we are able to relax some of the restrictive hypotheses that Feuer-Heymann [5], [6] have and also consider infinite dimensional control systems (distributed parameter systems). In the next section, we establish our notation and recall some basic definitions and facts from nonsmooth analysis and the theory of multifunctions. In Section 3 we study the problem of U(.)-invariance for nonlinear, finite dimensional control systems, extending the works of Feuer-Heymann [5], [6]. Finally, in Section 4 we address similar questions in the context of infinite dimensional, generally nonlinear, control systems.

Variational stability of infinite-dimensional optimal control problems

In this paper we examine the variational stability of infinite-dimensional optimal control problems governed by nonlinear evolution equations. Our tools are the Kuratowski—Mosco convergence of sets and the corresponding τ-convergence of functions. We prove the τ-convergence of cost functional and the convergence of the values of the problems, and we examine the variational stability of the solution and reachable sets. These results are then applied to a sequence of nonlinear parabolic distributed-parameter optimal control problems

PARETO OPTIMALITY FOR NONLINEAR INFINITE DIMENSIONAL CONTROL SYSTEMS

In this note we establish the existence of Pareto optimal solutions for nonlinear, infinite dimensional control systems with state dependent control constraints and an integral criterion taking values in a separable, reflexive Banach lattice. An example is also presented in detail. Our result extends earlier ones obtained by Cesari and Suryanarayana.

Infinite Horizon Optimal Control Problems for Semilinear Evolution Equations

In this paper we establish the existence of optimal solutions over an infinite planning horizon, of control systems governed by semilinear evolution equations and having time and state dependent control constraints. In particular, we show that every asymptotically minimizing sequence has a subsequence converging in some sense to the desired optimal pair