Nonlinear Parabolic Optimal Control Research Articles

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.

Optimal control of nonlinear evolution inclusions

In this paper, we study the optimal control of nonlinear evolution inclusions. First, we prove the existence of admissible trajectories and then we show that the set that they form is relatively sequentially compact and in certain cases sequentially compact in an appropriate function space. Then, with the help of a convexity hypothesis and using Cesari's approach, we solve a general Lagrange optimal control problem. After that, we drop the convexity hypothesis and pass to the relaxed system, for which we prove the existence of optimal controls, we show that it has a value equal to that of the original one, and also we prove that the original trajectories are dense in an appropriate topology to the relaxed ones. Finally, we present an example of a nonlinear parabolic optimal control that illustrates the applicability of our results.

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.

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.