Convergence Of Optimal Controls Research Articles

Instantaneous control of interacting particle systems in the mean-field limit

Controlling large particle systems in collective dynamics by a few agents is a subject of high practical importance, e.g., in evacuation dynamics. In this paper we study an instantaneous control approach to steer an interacting particle system into a certain spatial region by repulsive forces from a few external agents, which might be interpreted as shepherd dogs leading sheep to their home. We introduce an appropriate mathematical model and the corresponding optimization problem. In particular, we are interested in the interaction of numerous particles, which can be approximated by a mean-field equation. Due to the high-dimensional phase space this will require a tailored optimization strategy. The arising control problems are solved using adjoint information to compute the descent directions. Numerical results on the microscopic and the macroscopic level indicate the convergence of optimal controls and optimal states in the mean-field limit, i.e., for an increasing number of particles.

Convergence of optimal controls in some optimization problems

We consider the behavior of bundles of optimal controls when the initial state of the system goes to some given vector. We investigate three types of optimization problems: problems with fixed process length and a free right endpoint; problems with fixed process length and a fixed right endpoint; time-optimal problems. The article is a review of recent results obtained by the author.

Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems?Strong convergence of optimal controls

A class of optimal control problems for a parabolic equation with nonlinear boundary condition and constraints on the control and the state is considered. Associated approximate problems are established, where the equation of state is defined by a semidiscrete Ritz-Galerkin method. Moreover, we are able to allow for the discretization of admissible controls. We show the convergence of the approximate controls to the solution of the exact control problem, as the discretization parameter tends toward zero. This result holds true under the assumption of a certain sufficient second-order optimality condition.

Approximation of non-linear parabolic boundary control problems by the Fourier method - convergence of optimal control

Motivating the consideration of more general optimal control problems a boundary control problem for the one-dimensional heat equation with non-linear boundary condition is approximated by the Fourier method. The corresponding infinite series defining the state-function (obtained by the Fourier method) is terminated after n terms, and the set of admissible controls is replaced by a suitable approximation, too. It is shown that the sequence of “approximating” controls is strongly convergent in L 2 to a certain locally optimal control for n → ∞, provided that second order sufficient optimality conditions are fulfilled at this control and some additional assumptions on the non-linearities of the problem are satisfied. The theory is worked out for a general class of control problems governed by non-linear integral equations, which covers boundary control as a particular case.

On the approximate solution of time-optimal control problems

In this paper we are concerned with the approximate solution of time-optimal control problems in a nonreflexive Banach SpaceE by sequences of similar problems in Banach spacesEn which are assumed to approximateE in a fairly general sense. The problems under consideration are such that the solution operator of the associated evolution equation is a strongly continuous holomorphic contraction semigroup and the class of controls is taken from the dual of the Phillips adjoint space with respect to the infinitesimal generator of that semigroup. The main object is to establish convergence of optimal controls, transition times and corresponding trajectories of the approximating control problems which can be done by means of some results from the theory of approximation of semigroups of operators. Finally, these abstract convergence results will be applied to time-optimal control problems arising from heat transfer and diffusion processes.