54,136 research outputs found
Optimal control of the sweeping process over polyhedral controlled sets
The paper addresses a new class of optimal control problems governed by the
dissipative and discontinuous differential inclusion of the sweeping/Moreau
process while using controls to determine the best shape of moving convex
polyhedra in order to optimize the given Bolza-type functional, which depends
on control and state variables as well as their velocities. Besides the highly
non-Lipschitzian nature of the unbounded differential inclusion of the
controlled sweeping process, the optimal control problems under consideration
contain intrinsic state constraints of the inequality and equality types. All
of this creates serious challenges for deriving necessary optimality
conditions. We develop here the method of discrete approximations and combine
it with advanced tools of first-order and second-order variational analysis and
generalized differentiation. This approach allows us to establish constructive
necessary optimality conditions for local minimizers of the controlled sweeping
process expressed entirely in terms of the problem data under fairly
unrestrictive assumptions. As a by-product of the developed approach, we prove
the strong -convergence of optimal solutions of discrete
approximations to a given local minimizer of the continuous-time system and
derive necessary optimality conditions for the discrete counterparts. The
established necessary optimality conditions for the sweeping process are
illustrated by several examples
Design of a compensation for an ARMA model of a discrete time system
The design of an optimal dynamic compensator for a multivariable discrete time system is studied. Also the design of compensators to achieve minimum variance control strategies for single input single output systems is analyzed. In the first problem the initial conditions of the plant are random variables with known first and second order moments, and the cost is the expected value of the standard cost, quadratic in the states and controls. The compensator is based on the minimum order Luenberger observer and it is found optimally by minimizing a performance index. Necessary and sufficient conditions for optimality of the compensator are derived. The second problem is solved in three different ways; two of them working directly in the frequency domain and one working in the time domain. The first and second order moments of the initial conditions are irrelevant to the solution. Necessary and sufficient conditions are derived for the compensator to minimize the variance of the output
A pointwise tracking optimal control problem for the stationary Navier--Stokes equations
We study a pointwise tracking optimal control problem for the stationary
Navier--Stokes equations; control constraints are also considered. The problem
entails the minimization of a cost functional involving point evaluations of
the state velocity field, thus leading to an adjoint problem with a linear
combination of Dirac measures as a forcing term in the momentum equation, and
whose solution has reduced regularity properties. We analyze the existence of
optimal solutions and derive first and, necessary and sufficient, second order
optimality conditions in the framework of regular solutions for the
Navier--Stokes equations. We develop two discretization strategies: a
semidiscrete strategy in which the control variable is not discretized, and a
fully discrete scheme in which the control variable is discretized with
piecewise constant functions. For each solution technique, we analyze
convergence properties of discretizations and derive a priori error estimates
Stochastic optimal controls with delay
This thesis investigates stochastic optimal control problems with discrete delay and those with both discrete and exponential moving average delays, using the stochastic maximum principle, together with the methods of conjugate duality and dynamic programming.
To obtain the stochastic maximum principle, we first extend the conjugate duality method presented in [2, 44] to study a stochastic convex (primal) problem with discrete delay. An expression for the corresponding dual problem, as well as the necessary and sufficient conditions for optimality of both problems, are derived. The novelty of our work is that, after reformulating a stochastic optimal control problem with delay as a particular convex problem, the conditions for optimality of convex problems lead to the stochastic maximum principle for the control problem. In particular, if the control problem involves both the types of delay and is jump-free, the stochastic maximum principle obtained in this thesis improves those obtained in [29, 30].
Adapting the technique used in [19, Chapter 3] to the stochastic context, we consider a class of stochastic optimal control problems with delay where the value functions are separable, i.e. can be expressed in terms of so-called auxiliary functions. The technique enables us to obtain second-order partial differential equations, satisfied by the auxiliary functions, which we shall call auxiliary HJB equations. Also, the corresponding verification theorem is obtained. If both the types of delay are involved, our auxiliary HJB equations generalize the HJB equations obtained in [22, 23] and our verification theorem improves the stochastic verification theorem there
Analysis of the velocity tracking control problem for the 3D evolutionary Navier-Stokes equations
The velocity tracking problem for the evolutionary Navier–Stokes equations in three dimensions is studied. The controls are of distributed type and are submitted to bound constraints. The classical cost functional is modified so that a full analysis of the control problem is possible. First and second order necessary and sufficient optimality conditions are proved. A fully discrete scheme based on a discontinuous (in time) Galerkin approach, combined with conforming finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, Ď„ and h, respectively, satisfy Ď„ ≤ Ch2, the L2(ΩT ) error estimates of order O(h) are proved for the difference between the locally optimal controls and their discrete approximations. Finally, combining these techniques and the approach of Casas, Herzog, and Wachsmuth [SIAM J. Optim., 22 (2012), pp. 795–820], we extend our results to the case of L1(ΩT ) type functionals that allow sparse controls.This author was partially supported by the Spanish Ministerio de EconomĂa y
Competitividad under projects MTM2011-22711 and MTM2014-57531-
Stochastic optimal controls with delay
This thesis investigates stochastic optimal control problems with discrete delay and those with both discrete and exponential moving average delays, using the stochastic maximum principle, together with the methods of conjugate duality and dynamic programming.
To obtain the stochastic maximum principle, we first extend the conjugate duality method presented in [2, 44] to study a stochastic convex (primal) problem with discrete delay. An expression for the corresponding dual problem, as well as the necessary and sufficient conditions for optimality of both problems, are derived. The novelty of our work is that, after reformulating a stochastic optimal control problem with delay as a particular convex problem, the conditions for optimality of convex problems lead to the stochastic maximum principle for the control problem. In particular, if the control problem involves both the types of delay and is jump-free, the stochastic maximum principle obtained in this thesis improves those obtained in [29, 30].
Adapting the technique used in [19, Chapter 3] to the stochastic context, we consider a class of stochastic optimal control problems with delay where the value functions are separable, i.e. can be expressed in terms of so-called auxiliary functions. The technique enables us to obtain second-order partial differential equations, satisfied by the auxiliary functions, which we shall call auxiliary HJB equations. Also, the corresponding verification theorem is obtained. If both the types of delay are involved, our auxiliary HJB equations generalize the HJB equations obtained in [22, 23] and our verification theorem improves the stochastic verification theorem there
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