1,720,977 research outputs found
A stochastic gradient method for a class of nonlinear PDE-constrained optimal control problems under uncertainty
Full text linkThe study of optimal control problems under uncertainty plays an important
role in scientific numerical simulations. This class of optimization problems
is strongly utilized in engineering, biology and finance. In this paper, a
stochastic gradient method is proposed for the numerical resolution of a
nonconvex stochastic optimization problem on a Hilbert space. We show that,
under suitable assumptions, strong or weak accumulation points of the iterates
produced by the method converge almost surely to stationary points of the
original optimization problem. Measurability and convergence rates of a
stationarity measure are handled, filling a gap for applications to nonconvex
infinite dimensional stochastic optimization problems. The method is
demonstrated on an optimal control problem constrained by a class of elliptic
semilinear partial differential equations (PDEs) under uncertainty
On the metric regularity of affine optimal control problems
No full textThe paper establishes properties of the type of (strong) metric regularity of the set-valued map associated with the system of necessary optimality conditions for optimal control problems that are affine with respect to the control (shortly, affine problems). It is shown that for such problems it is reasonable to extend the standard notions of metric regularity by involving two metrics in the image space of the map. This is done by introducing (following an earlier paper by the first and the third named author) the concept of (strong) bi-metric regularity in a general space setting. Lyusternik-Graves-type theorems are proved for (strongly) bi-metrically regular maps, which claim stability of these regularity properties with respect to “appropriately small” perturbations. Based on that, it is proved that in the case of a map associated with affine optimal control problems, the strong bi-metric regularity is invariant with respect to linearization. This result is complemented with a sufficient condition for strong bi-metric regularity for linear-quadratic affine optimal control problems, which applies to the “linearization” of a nonlinear affine problem. Thus the same conditions are also sufficient for strong bi-metric regularity in the nonlinear affine problem
Higher-order numerical scheme for linear quadratic problems with bang–bang controls
Full text linkThis paper considers a linear-quadratic optimal control problem where the control function appears linearly and takes values in a hypercube. It is assumed that the optimal controls are of purely bang-bang type and that the switching function, associated with the problem, exhibits a suitable growth around its zeros. The authors introduce a scheme for the discretization of the problem that doubles the rate of convergence of the Euler's scheme. The proof of the accuracy estimate employs some recently obtained results concerning the stability of the optimal solutions with respect to disturbances
Sensitivity relations for the Mayer problem of optimal control
No full textSensitivity relations in optimal control refer to the interpretation of the gradients of the value function in terms of the costate arc and the Hamiltonian evaluated along an extremal. In general, the value function is not differentiable and for this reason its gradients have to be replaced by generalized differentials. In this paper we prove such sensitivity relations for the Mayer optimal control problem with dynamics described by a differential inclusion. If the associated Hamiltonian is semiconvex with respect to the state variable, then we show that sensitivity relations hold true for any dual arc associated to an optimal solution, instead of more traditional statements about the existence of a dual arc satisfying such relations. Furthermore, several applications are provided
Second-order sensitivity relations and regularity of the value function for mayer's problem in optimal control
No full textThis paper investigates the value function, V, of a Mayer optimal control problem with the state equation given by a differential inclusion. First, we obtain an invariance property for the proximal and Fŕechet subdifferentials of V along optimal trajectories. Then, we extend the analysis to the sub-and superjets of V, obtaining new sensitivity relations of second order. By applying sensitivity analysis to exclude the presence of conjugate points, we deduce that the value function is twice differentiable along any optimal trajectory starting at a point at which V is proximally subdifferentiable. We also provide sufficient conditions for the local C2 regularity of V on neighborhoods of optimal trajectories
Stochastic Proximal Gradient Methods for Nonconvex Problems in Hilbert Spaces
Full text linkFor finite-dimensional problems, stochastic approximation methods have long been used to solve stochastic optimization problems. Their application to infinite-dimensional problems is less understood, particularly for nonconvex objectives. This paper presents convergence results for the stochastic proximal gradient method applied to Hilbert spaces, motivated by optimization problems with partial differential equation (PDE) constraints with random inputs and coefficients. We study stochastic algorithms for nonconvex and nonsmooth problems, where the nonsmooth part is convex and the nonconvex part is the expectation, which is assumed to have a Lipschitz continuous gradient. The optimization variable is an element of a Hilbert space. We show almost sure convergence of strong limit points of the random sequence generated by the algorithm to stationary points. We demonstrate the stochastic proximal gradient algorithm on a tracking-type functional with a L1-penalty term constrained by a semilinear PDE and box constraints, where input terms and coefficients are subject to uncertainty. We verify conditions for ensuring convergence of the algorithm and show a simulation
Sensitivity relations for the Mayer problem with differential inclusions
No full textIn optimal control, sensitivity relations are usually understood as inclusions that identify the pair formed by the dual arc and the Hamiltonian as a suitable generalized gradient of the value function, evaluated along a given minimizing trajectory. In this paper, sensitivity relations are obtained for the Mayer problem associated with the differential inclusion F(x) and applied to express optimality conditions. The first application of our results concerns the maximum principle and consists in showing that a dual arc can be constructed for every element of the superdifferential of the final cost as a solution of an adjoint system. The second and last application we discuss in this paper concerns optimal design. We show that one can associate a family of optimal trajectories, starting at some point (t,x), with every nonzero reachable gradient of the value function at (t,x), in such a way that families corresponding to distinct reachable gradients have empty intersection
Regularity results for the minimum time function with Hörmander vector fields
Full text linkIn a bounded domain of Rnwith boundary given by a smooth (n −1)-dimensional manifold, we consider the homogeneous Dirichlet problem for the eikonal equation associated with a family of smooth vector fields {X1, ..., XN} subject to Hörmander’s bracket generating condition. We investigate the regularity of the viscosity solution Tof such problem. Due to the presence of characteristic boundary points, singular trajectories may occur. First, we characterize these trajectories as the closed set of all points at which the solution loses point-wise Lipschitz continuity. Then, we prove that the local Lipschitz continuity of T, the local semiconcavity of T, and the absence of singular trajectories are equivalent properties. Finally, we show that the last condition is satisfied whenever the characteristic set of {X1, ..., XN}is a symplectic manifold. We apply our results to several examples
Metric Regularity Properties in Bang-Bang Type Linear-Quadratic Optimal Control Problems
Full text linkThe paper investigates the Lipschitz/Hölder stability with respect to perturbations of the solutions of linear-quadratic optimal control problems where the control variable appears linearly and the optimal one is of bang-bang type. Conditions for bi-metric regularity and (Hölder) metric sub-regularity are established, involving only the order of the zeros of the associated switching function and smoothness of the data. The results provide a basis for investigation of various approximation methods and are applied in this paper for convergence analysis of a Newton-type method
High Order Discrete Approximations to Mayer's Problems for Linear Systems
No full textThis paper presents a discretization scheme for Mayer's type optimal control problems of linear systems. The scheme is based on second order Volterra--Fliess approximations, and on an augmentation of the control variable in a control set of higher dimension. Compared with the existing results, it has the advantage of providing a higher order accuracy, which may make it more efficient when aiming for a certain precision. Error estimations (depending on the controllability index of the system at the solution) are proved by using a recent result about stability of the optimal solution with respect to disturbances. Numerical results are provided which show the sharpness of the error estimations.
Read More: http://epubs.siam.org/doi/abs/10.1137/16M107914
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