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In this paper we discuss the optimal control of a quasilinear parabolic state equation. Its form is leaned on the kind of problems arising for example when controlling the anisotropic Allen–Cahn equation as a model for crystal growth. Motivated by this application we consider the state equation as a result of a gradient flow of an energy functional. The quasilinear term is strongly monotone and obeys a certain growth condition and the lower order term is non-monotone. The state equation is discretized implicitly in time with piecewise constant functions. The existence of the control-to-state operator and its Lipschitz-continuity is shown for the time discretized as well as for the time continuous problem. Latter is based on the convergence proof of the discretized solutions. Finally we present for both the existence of global minimizers. Also convergence of a subsequence of time discrete optimal controls to a global minimizer of the time continuous problem can be shown. Our results hold in arbitrary space dimensions. [ABSTRACT FROM AUTHOR]

This paper is concerned with a bilinear control problem for enhancing convection-cooling via an incompressible velocity field. Both optimal open-loop control and closed-loop feedback control designs are addressed. First and second order optimality conditions for characterizing the optimal solution are discussed. In particular, the method of instantaneous control is applied to establish the feedback laws. Moreover, the construction of feedback laws is also investigated by directly utilizing the optimality system with appropriate numerical discretization schemes. Computationally, it is much easier to implement the closed-loop feedback control than the optimal open-loop control, as the latter requires to solve the state equations forward in time, coupled with the adjoint equations backward in time together with a nonlinear optimality condition. Rigorous analysis and numerical experiments are presented to demonstrate our ideas and validate the efficacy of the control designs. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a class of coupled multidimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter H ∈ (1 / 2 , 1) . Using pathwise approach and based on Perov's fixed point theorem, we prove that the existence and uniqueness of solution for the equations considered under some local Lipschitz conditions. Subsequently, by establishing some priori estimates, we obtain a viability result for the stochastic systems under investigation. The sufficient and necessary condition is also an alternative global existence result for the fractional differential equations with restrictions on the state. Finally, by direct and inverse images for fractional stochastic tangent sets, we establish the deterministic necessary and sufficient conditions which control that the solution for the coupled stochastic systems under investigation evolves in some particular sets. [ABSTRACT FROM AUTHOR]