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A self-learning approach for optimal feedback gains for finite-horizon nonlinear continuous time control systems is proposed and analysed. It relies on parameter dependent approximations to the optimal value function obtained from a family of universal approximators. The cost functional for the training of an approximate optimal feedback law incorporates two main features. First, it contains the average over the objective functional values of the parametrized feedback control for an ensemble of initial values. Second, it is adapted to exploit the relationship between the maximum principle and dynamic programming. Based on universal approximation properties, existence, convergence and first order optimality conditions for optimal neural network feedback controllers are proved.

Stabilization of a class of time-varying parabolic equations with uncertain input data using receding horizon control (RHC) is investigated. The diffusion coefficient and the initial function are prescribed as random fields. We consider both cases: uniform and log-normal distributions of the diffusion coefficient. The controls are chosen to be finite-dimensional and enter into the system as a linear combination of finitely many indicator functions (actuators) supported in open subsets of the spatial domain. Under suitable regularity assumptions, we study the expected (averaged) stabilizability of the RHC-controlled system with respect to the number of actuators. An upper bound is also obtained for the failure probability of RHC in relation to the choice of the number of actuators and parameters in the equation. [ABSTRACT FROM AUTHOR]

The stabilizability to trajectories of the Schlögl model is investigated in the norm of the natural state space for strong solutions, which is strictly contained in the standard pivot space of square integrable functions. As actuators a finite number of indicator functions are used and the control input is subject to a bound constraint. A stabilizing saturated explicit feedback control is proposed, where the set of actuators and the input bound are independent of the targeted trajectory. Further, the existence of open-loop optimal stabilizing constrained controls and related first-order optimality conditions are investigated. These conditions are then used to compute stabilizing receding horizon based controls. Results of numerical simulations are presented comparing their stabilizing performance with that of saturated explicit feedback controls. [ABSTRACT FROM AUTHOR]

The long time behavior and detailed convergence analysis of Langevin equations have received increased attention in recent years. Difficulties arise from a lack of coercivity, usually termed hypocoercivity, of the underlying kinetic Fokker-Planck operator, which is a consequence of the partially deterministic nature of a second order stochastic differential equation. In this manuscript, the effect of controlling the confinement potential without altering the original invariant measure is investigated. This leads to an abstract bilinear control system with an unbounded but infinite-time admissible control operator which, by means of an artificial diffusion approach, is shown to possess a unique solution. The compactness of the underlying semigroup is further used to define an infinite-horizon optimal control problem on an appropriately reduced state space. Under smallness assumptions on the initial data, feasibility of and existence of a solution to the optimal control problem are discussed. Numerical results based on a local approximation based on a shifted Riccati equation illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

This is the continuation of our work on infinite horizon optimal control problems with a discount factor on the state variable and nonlinear partial differential equations as constraints. Existence of a solution is proven, and first as well as second order optimality conditions are derived. They are used to analyze the approximation of the infinite horizon problem by finite horizon problems. [ABSTRACT FROM AUTHOR]

This is the first part of an investigation of infinite horizon optimal control problems subject to semilinear parabolic equations. A discount factor on the state variable is introduced in the cost. This allows for the treatment of infinite horizon problems without stabilizability assumptions. The nonlinearities can be of polynomial type, thus covering reaction-diffusion equations which are important for applications. The control-to-state mapping and its regularity are analyzed in detail. This involves the relation between the type of the nonlinearity and the discount factor. [ABSTRACT FROM AUTHOR]

The numerical approximation of an optimal control problem governed by a semilinear parabolic equation and constrained by a bound on the spatial |$L^1$| -norm of the control at every instant of time is studied. Spatial discretizations of the controls by piecewise constant and continuous piecewise linear functions are investigated. Under finite element approximations, the sparsity properties of the continuous solutions are preserved in a natural way using piecewise constant approximations of the control, but suitable numerical integration of the objective functional and of the constraint must be used to keep the sparsity pattern when using spatially continuous piecewise linear approximations. We also obtain error estimates and finally present some numerical examples. [ABSTRACT FROM AUTHOR]