Optimal control in nonlinear infinite-dimensional systems with nondifferentiability of two types

We consider the optimal control problem for systems described by nonlinear equations of elliptic type. If the nonlinear term in the equation is smooth and the nonlinearity increases at a comparatively low rate of growth, then necessary conditions for optimality can be obtained by well-known methods. For small values of the nonlinearity exponent in the smooth case, we propose to approximate the state operator by a certain differentiable operator. We show that the solution of the approximate problem obtained by standard methods ensures that the optimality criterion for the initial problem is close to its minimal value. For sufficiently large values of the nonlinearity exponent, the dependence of the state function on the control is nondifferentiable even under smoothness conditions for the operator. But this dependence becomes differentiable in a certain extended sense, which is sufficient for obtaining necessary conditions for optimality. Finally, if there is no smoothness and no restrictions are imposed on the nonlinearity exponent of the equation, then a smooth approximation of the state operator is possible. Next, we obtain necessary conditions for optimality of the approximate problem using the notion of extended differentiability of the solution of the equation approximated with respect to the control, and then we show that the optimal control of the approximated extremum problem minimizes the original functional with arbitrary accuracy.     

An optimal control problem for a nonlinear elliptic equation with a phase constraint and state variation

We study an optimal control problem for a system described by a nonlinear elliptic equation with a state constraint in the form of an inclusion. We prove the solvability of the problem under consideration and by varying the state of the system obtain necessary optimality conditions.     

Approximation methods in optimal control problems for nonlinear infinite-dimensional systems

Some notions related to approximate solutions and to the approximation of extremum problems for nonlinear infinite-dimensional systems are proposed. Optimization problems for nonlinear parabolic equations with a fixed terminal state and on an infinite time interval, as well as for singular stationary systems with phase constraints, are illustrated by several examples.     

Optimal control of nonlinear evolution systems in the case where the solution is not differentiable with respect to the control

For the simplest heat equation with power nonlinearity, the dependence of the solution of the corresponding boundary-value problem on the constant term of the equation turns out to be, in general, not differentiable in the sense of Gâteaux.     

Approximate Solution of Singular Optimization Problems

We study the optimal control problem for systems described by nonlinear elliptic equations. We have no information about the existence and uniqueness of the solution for some particular control. The extremum problem may be unsolvable. We regularize the problem by using a combination of the penalty method and the Tikhonov method. For the regularized problem, we prove the existence of the solution and find necessary conditions for optimality in the form of variational inequalities. We show that the regularization method used in this paper allows one to find an approximate (in some sense) solution of the original problem.     

Approximate Solution of Optimization Problems for Infinite-Dimensional Singular Systems

An optimal control problem is considered for a system described by a singular equation of parabolic type. The study bases on a special regularization method. We establish existence of a solution to the regularized problem, as well as the corresponding necessary optimality conditions. The results enable us to find an approximate solution to the original problem even in the absence of solvability.     

The necessary optimality conditions for a nonlinear stationary system whose state functional is not differentiable with respect to the control

We consider a control system described by a nonlinear elliptic equation. Its control-state mapping is extendedly differentiable but not Gateaux differentiable for large values of the domain dimension and the nonlinearity index. We obtain the necessary optimality condition for various state functionals.