Optimal control of nonlinear evolution equations with nonmonotone nonlinearities

In this paper, we prove the existence of optimal admissible pairs for a large class of strongly nonlinear evolution equations, involving nonmonotone nonlinearities. An example of a nonlinear parabolic optimal control system is also worked out in detail.The author wishes to thank Professor T. S. Angell for useful comments and suggestions.This research was supported by NSF Grant No. DMS-88-02688.     

Approximation of relaxed nonlinear parabolic optimal control problems

We consider a relaxed optimal control problem for systems defined by nonlinear parabolic partial differential equations with distributed control. The problem is completely discretized by using a finite-element approximation scheme with piecewise linear states and piecewise constant controls. Existence of optimal controls and necessary conditions for optimality are derived for both the continuous and the discrete problem. We then prove that accumulation points of sequences of discrete optimal [resp. extremal] controls are optimal [resp. extremal] for the continuous problem.     

Properties of the solution set of nonlinear evolution inclusions

In this paper we examine nonlinear, nonautonomous evolution inclusions defined on a Gelfand triple of spaces. First we show that the problem with a convex-valued,h*-usc inx orientor fieldF(t, x) has a solution set which is anR _δ-set inC(T, H). Then for the problem with a nonconvex-valuedF(t, x) which ish-Lipschitz inx, we show that the solution set is path-connected inC(T, H). Subsequently we prove a strong invariance result and a continuity result for the solution multifunction. Combining these two results we establish the existence of periodic solutions. Some examples of parabolic partial differential equations with multivalued terms are also included. This work was done while the authors were visiting the Florida Institute of Technology.     

Optimal control of infinite-dimensional uncertain systems

In this paper, we consider a minimax problem of optimal control for a class of strongly nonlinear uncertain evolution equations on a Banach space. We prove the existence of optimal controls. A nontrivial example of a class of systems governed by a nonlinear partial differential equation with uncertain spatial parameters is presented for illustration.This work was supported in part by the National Science and Engineering Research Council of Canada under Grant No. A7109 and The Engineering Faculty Development Fund, University of Ottawa.The authors would like to thank the anonymous reviewers for their valuable comments and suggestions.     

Optimal control problems for a class of nonlinear evolution equations

In this paper we study optimal control problems for systems governed by nonlinear evolution equations. First we develop an existence theory for systems with a priori feedback using the reduction technique and a convexity-type hypothesis involving property Q. In doing this we also establish the nonemptiness of the set of admissible state-control pairs, by solving a nonlinear evolution inclusion. Then we obtain necessary conditions for optimality for a class of problems with terminal cost criterion and initial condition which is not a priori given but is only required to belong to a given set (systems with insufficient data in the terminology of Lions). This revised version was published online in June 2006 with corrections to the Cover Date.     

Sensitivity analysis of distributed-parameter optimal control problems for nonlinear parabolic equations

A sequence of optimal control problems for systems described by nonlinear parabolic equations is considered. It is proved that, under the -convergence of objective functionals, the parabolicG-convergence of operators in the state equations, and the Kuratowski convergence of control constraint sets, a convergent sequence of optimal pairs has a limit which is an optimal pair for the limit control problem. The convergence of minimal values is also obtained.This research was supported in part by the Istituto Nazionale di Alta Matematica F. Severi, Rome, Italy. Part of this research was carried out while the author was visiting the Scuola Normale Superiore, Pisa, Italy.     

Optimal control of parabolic equation with an infinite number of variables for non-standard functional and time delay

A distributed control problem for the parabolic operator withan infinite number of variables and time delay is considered.The performance index has an integral form. Constraints on controlsare imposed. To obtain optimality conditions for the Neumannproblem, the generalization of the Dubovitskii–Milyutintheorem given by Walczak in WALCZAK, S. Folia Mathematics, 1,187–196 and WALCZAK, S. J. Optim. Theory Appl., 42, 561–582was applied.     

Asymptotic behavior of optimal solutions to control problems for systems described by differential inclusions corresponding to partial differential equations

In the paper, we consider differential inclusions related to PDEs of parabolic type and some control problems with integral cost functionals associated to them. Given a sequence of such problems, we investigate first the asymptotic behavior of solution sets (mild solutions or more precisely selection-trajectory pairs) for differential inclusions, and we get some semicontinuity or continuity results (Kuratowski convergence of solution sets). Then, we prove the -convergence of cost functionals, related to the above Kuratowski convergence of solution sets. Finally, applying the Buttazzo-Dal Maso abstract scheme, based on the sequential -convergence, we obtain results concerning the asymptotic behavior (hence, also stability results) for optimal solutions to control problems as well as the convergence of minimal values.The authors would like to thank Professors G. Dal Maso and S. Spagnolo for helpful conversations.This work was done when the first author was visiting ICTP and ISAS in Trieste in 1990/91.