On Parametric Evolution Inclusions of the Subdifferential Type with Applications to Optimal Control Problems

Abstract

Abstract. In this paper we study parametric evolution inclusions of the subdifferential type and their applications to the sensitivity analysis of nonlinear, infinite dimensional optimal control problems. The parameter appears in all the data of the problem, including the subdifferential operator. First we establish several continuity results for the solution multifunction of the subdifferential inclusion. Then we study how these results can be used to examine the sensitivity properties (variational stability) of certain broad classes of nonlinear infinite dimensional optimal control problems. Some examples are worked out in detail, illustrating the applicability of our work. These include obstacle problems (with time varying obstacles), optimal control of distributed parameter systems, and differential variational inequalities.

Optimal Control Problems for Evolutionary Variational Inequalities with Volterra type Operators

“…Also, optimal control problems for some variational and hemivariational inequalities were considered in [12] and [13], respectively. The optimal control problems for the subdifferential evolution inclusions have been examined in many works, see, e.g., [14], [15], [16] and references therein. More precisely, in [15] it was studied the Volterra integrodifferential evolution inclusions of nonconvolution type with time dependent unbounded operators and with both convex and nonconvex multivalued perturbations.…”

Multiple Solutions with Sign Information for a Class of Parametric Superlinear (p, 2)-Equations

“…Finally such sensitivity analysis for parametric equations is also important in the study of optimization and control problems. It provides information about the tolerance of the systems on the variation of the parameter and in which range we expect to find optimal solutions (see Papageorgiou [22,23] and Sokołowski [32]).…”

Sensitivity analysis for optimal control problems governed by nonlinear evolution inclusions

“…Such a sensitivity analysis (also known in the literature as "variational analysis") is important because it gives information about the tolerances which are permitted in the specification of the mathematical models, it suggests ways to solve parametric problems and also can be useful in the computational analysis of the problem. For infinite dimensional systems (distributed parameter systems), such investigations were conducted by Buttazzo and Dal Maso [8], Denkowski and Migorski [13], Ito and Kunisch [24], Papageorgiou [31] (linear systems), Papageorgiou [30], Sokolowski [38] (semilinear systems) and Hu and Papageorgiou [23], Papageorgiou [32,33] (nonlinear systems). We also mention the books of Buttazzo [7], Dontchev and Zolezzi [17], Ito and Kunisch [25], Sokolowski and Zolezio [39] (the latter for shape optimization problems).…”