On second order necessary conditions in infinite dimensional optimal control with state constraints

This paper is devoted to second order necessary optimality conditions for control problems in infinite dimensions. The main novelty of our work is the presence of pure state constraints together with end point constraints, quite useful in the applications. Second order analysis for control problems involving PDEs has been extensively discussed in the literature. The most usual approach to derive necessary optimality conditions is to rewrite the control problem as an abstract mathematical programming one. Our approach is different, we avoid the reformulation of the optimal control problem and use instead second order variational analysis. The necessary optimality conditions are in the form of a maximum principle and a second order variational inequality. They are first obtained in the form of nonintersection of convex sets. A suitable separation theorem allows to deduce their dual characterization

Optimal control of Allen-Cahn systems

Optimization problems governed by Allen-Cahn systems including elastic effects are formulated and first-order necessary optimality conditions are presented. Smooth as well as obstacle potentials are considered, where the latter leads to an MPEC. Numerically, for smooth potential the problem is solved efficiently by the Trust-Region-Newton-Steihaug-cg method. In case of an obstacle potential first numerical results are presented

A sufficient optimality condition for delayed state-linear optimal control problems

We give answer to an open question by proving a sufficient optimality condition for state-linear optimal control problems with time delays in state and control variables. In the proof of our main result, we transform a delayed state-linear optimal control problem to an equivalent non-delayed problem. This allows us to use a well-known theorem that ensures a sufficient optimality condition for non-delayed state-linear optimal control problems. An example is given in order to illustrate the obtained result.Comment: This is a preprint of a paper whose final and definite form is with 'Discrete and Continuous Dynamical Systems -- Series B' (DCDS-B), ISSN 1531-3492, eISSN 1553-524X, available at [http://www.aimsciences.org/journal/1531-3492]. Paper Submitted 31/Dec/2017; Revised 13/April/2018; Accepted 11/Jan/201

High order variational integrators in the optimal control of mechanical systems

In recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class are variational integrators which are momentum preserving and symplectic. In this article, we develop two high order variational integrators which distinguish themselves in the dimension of the underling space of approximation and we investigate their application to finite-dimensional optimal control problems posed with mechanical systems. The convergence of state and control variables of the approximated problem is shown. Furthermore, by analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that, for these particular integrators, dualization and discretization commute.Comment: 25 pages, 9 figures, 1 table, submitted to DCDS-

On optimal solution error covariances in variational data assimilation problems

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find unknown parameters such as distributed model coefficients or boundary conditions. The equation for the optimal solution error is derived through the errors of the input data (background and observation errors), and the optimal solution error covariance operator through the input data error covariance operators, respectively. The quasi-Newton BFGS algorithm is adapted to construct the covariance matrix of the optimal solution error using the inverse Hessian of an auxiliary data assimilation problem based on the tangent linear model constraints. Preconditioning is applied to reduce the number of iterations required by the BFGS algorithm to build a quasi-Newton approximation of the inverse Hessian. Numerical examples are presented for the one-dimensional convection-diffusion model

A FEM for an optimal control problem of fractional powers of elliptic operators

We study solution techniques for a linear-quadratic optimal control problem involving fractional powers of elliptic operators. These fractional operators can be realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder in one more spatial dimension. Thus, we consider an equivalent formulation with a nonuniformly elliptic operator as state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We discretize the proposed truncated state equation using first degree tensor product finite elements on anisotropic meshes. For the control problem we analyze two approaches: one that is semi-discrete based on the so-called variational approach, where the control is not discretized, and the other one is fully discrete via the discretization of the control by piecewise constant functions. For both approaches, we derive a priori error estimates with respect to the degrees of freedom. Numerical experiments validate the derived error estimates and reveal a competitive performance of anisotropic over quasi-uniform refinement