Optimal Control Problems For Some Partial Differential Equations

In this thesis, we mainly study the optimal control problems for some partial differential equations, which mainly contains the well-posedness of the controlled system, the existence of optimal control and the maximum principle for optimal control. The thesis consists of six chapters:Chapter One summarizes the background of the optimal control problems for PDEs in the past decades. We also state the main results and give the notations used in the whole thesis.Chapter Two considers the optimal distributed control problems for a class of nonlin-ear dispersion equation which involves the Novikov equation. First, by choosing a suitable working space and using the energy method, we prove that the controlled system is locally well-posed. Second, we introduce the optimal control problem that we take into consid-eration, and prove that the controlled system admits a unique optimal control. Finally, by means of the method suggested by A.Ya. Dubovitskii and A.A. Milyutin, we establish the first order necessary optimality condition of optimal control for the controlled system in fixed final horizon case.Chapter Three is devoted to the optimal control problem for an ecosystem composed of one predator and two competing preys, and our goal is to maximize the total density of the three populations. By using the strongly continuous semigroup theory, We first investigate the existence and uniqueness of the positive strong solution for the controlled system, and find an optimal solution under given initial conditions. Then we establish the first order necessary optimality condition, and point out that the optimal control has a bang-bang form. Moreover, the second order necessary and sufficient optimality conditions are established.Chapter Four deals with a the optimal control problems governed by coupled nonlinear wave equations with memory. First, by using the classical variational principle and the com-pact principle, we investigate the existence of the optimal solution for the optimal control problem. Second, and then we deduce the maximum principle for the optimal solution with state constraint. Moreover, we give a concrete example which is covered by the main result.Chapter Five studies the state-constrained optimal control problems governed by the Boussinesq equations with pointwise state constraint. We first prove that the state variable continuously depend on the control variable, then we obtain a Pontryagin's maximum prin-ciple (pointwise type) for optimal control by using the Ekeland's variational principle and a revised type of spike perturbation. In addition, a type of 'strong' Pontryagin maximum principle under suitable stability conditions is also derived, the Lagrange multiplier take the fixed value 1.Chapter Six considers the local well-posedness, blow-up phenomena and analytical solution for a class of shallow water wave equations with (k+1)-order nonlinearity. First, by means of the the Littlewood-Paley theory and the transport equation theory, we establish the local well-posedness of the equation in Bp,rs. Second, the local well-posedness in the critical space is also obtained. Third, we derive the blow-up criteria and prove the global existence of strong solution in a special case. Finally, we investigate the Gevrey regularity and analyticity of the solutions, and we get a lower bound of the lifespan and the continuity of the data-to-solution mapping.