Scheme For Optimal Control Problems Research Articles

Implicit Runge–Kutta Schemes for Optimal Control Problems with Evolution Equations

Abstract In this paper we discuss the use of implicit Runge–Kutta schemes for the time discretization of optimal control problems with evolution equations. The specialty of the considered discretizations is that the discretizations schemes for the state and adjoint state are chosen such that discretization and optimization commute. It is well known that for Runge–Kutta schemes with this property additional order conditions are necessary. We give sufficient conditions for which class of schemes these additional order condition are automatically fulfilled. The focus is especially on implicit Runge–Kutta schemes of Gauss, Radau IA, Radau IIA, Lobatto IIIA, Lobatto IIIB and Lobatto IIIC collocation type up to order six. Furthermore, we also use a SDIRK (singly diagonally implicit Runge–Kutta) method to demonstrate, that for general implicit Runge–Kutta methods the additional order conditions are not automatically fulfilled. Numerical examples illustrate the predicted convergence rates.

Nonlinear optimal control: a numerical scheme based on occupation measures and interval analysis

This paper presents an approximation scheme for optimal control problems using finite-dimensional linear programs and interval analysis. This is done in two parts. Following Vinter approach (SIAM J Control Optim 31(2):518–538, 1993) and using occupation measures, the optimal control problem is written into a linear programming problem of infinite-dimension (weak formulation). Thanks to Interval arithmetic, we provide a relaxation of this infinite-dimensional linear programming problem by a finite dimensional linear programming problem. A proof that the optimal value of the finite dimensional linear programming problem is a lower bound to the optimal value of the control problem is given. Moreover, according to the fineness of the discretization and the size of the chosen test function family, obtained optimal values of each finite dimensional linear programming problem form a sequence of lower bounds which converges to the optimal value of the initial optimal control problem. Examples will illustrate the principle of the methodology.

Lebesgue constants arising in a class of collocation methods

Estimates are obtained for the Lebesgue constants associated with the Gauss quadrature points on $(-1, +1)$ augmented by the point $-1$ and with the Radau quadrature points on either $(-1, +1]$ or $[-1, +1)$. It is shown that the Lebesgue constants are $O(\sqrt{N})$, where $N$ is the number of quadrature points. These point sets arise in the estimation of the residual associated with recently developed orthogonal collocation schemes for optimal control problems. For problems with smooth solutions, the estimates for the Lebesgue constants can imply an exponential decay of the residual in the collocated problem as a function of the number of quadrature points.

A two-grid discretization scheme for optimal control problems of elliptic equations

In the paper, a two-grid finite element scheme is discussed for distributed optimal control governed by elliptic equations. With this new scheme, the solution of the elliptic optimal control problem on a fine grid is reduced to the solution of the elliptic optimal control problem on a much coarser grid and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. Finally, numerical experiments are carried out to confirm the considered theory.

Symmetric error estimates for discontinuous Galerkin time-stepping schemes for optimal control problems constrained to evolutionary Stokes equations

We consider fully discrete finite element approximations of a distributed optimal control problem, constrained by the evolutionary Stokes equations. Conforming finite element methods for spatial discretization combined with discontinuous time-stepping Galerkin schemes are being used for the space-time discretization. Error estimates are proved under weak regularity hypotheses for the state, adjoint and control variables. The estimates are also applicable when high order schemes are being used. Computational examples validating our expected rates of convergence are also provided.

Asymptotic preserving time‐discretization of optimal control problems for the Goldstein–Taylor model

We consider the development of implicit‐explicit time integration schemes for optimal control problems governed by the Goldstein–Taylor model. In the diffusive scaling, this model is a hyperbolic approximation to the heat equation. We investigate the relation of time integration schemes and the formal Chapman–Enskog‐type limiting procedure. For the class of stiffly accurate implicit–explicit Runge–Kutta methods, the discrete optimality system also provides a stable numerical method for optimal control problems governed by the heat equation. Numerical examples illustrate the expected behavior. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1770–1784, 2014

Discrete time schemes for optimal control problems with monotone controls

In this article, we consider the Hamilton–Jacobi–Bellman equation associated with the optimization problem with monotone controls. The problem deals with the infinite horizon case and costs with update coefficients. We study the numerical solution through the discretization in time by finite differences. Without the classical semiconcavity-like assumptions, we prove that the convergence in this problem is of order $$h^\gamma $$ in contrast with the order $$h^{\frac{\gamma }{2}}$$ valid for general control problems. This difference arises from the simple and precise way the monotone controls can be approximated. We illustrate the result with a simple example.

A convergence computational scheme for optimal control problems governed by Schrödinger equations

In this paper, a numerical scheme for obtaining approximate solutions of optimal control problems governed by Schrodinger equations is presented. In this method, by considering a partition of the control space, discrete form of the problem is converted to a quasi assignment problem. Then using an evolutionary algorithm, approximate optimal control is obtained as a piecewise linear function. The comparison between the numerical solution and the exact solution for the test cases shows the good accuracy of the presented method.

A mass-conservative characteristic FE scheme for optimal control problems governed by convection–diffusion equations

In this paper, we develop a mass-conservative characteristic finite element scheme for optimal control problems governed by linear singularly perturbed, convection–diffusion equations. We discuss the case that the velocity fields are non-divergence-free. The space discretization of the state variable is done by piecewise linear continuous functions, whereas the control variable is approximated by piecewise constant functions due to the limitation of the regularity. The scheme preserves the mass balance for the original state equation. We derive a priori error estimates for both the control and state approximations. Some numerical examples are presented to show the efficiency of the proposed scheme.

Crank--Nicolson Schemes for Optimal Control Problems with Evolution Equations

Crank--Nicolson methods are often used for the simulation of initial boundary value problems for parabolic partial differential equations. In this paper we present a family of discretizations for parabolic optimal control problems based on Crank--Nicolson schemes with different time discretizations for state $y$ and adjoint state $p$ so that discretization and optimization commute. One of these methods can also be explained as a Störmer--Verlet scheme in the context of geometric numerical integration of Hamiltonian systems. Two of the schemes may also be obtained as a Galerkin method with quadrature. Further we investigate the schemes for a variable time step size and prove second order convergence for this case if the time step size is chosen with respect to an arbitrary, sufficiently smooth mesh generating function.

Time discretizations for numerical optimisation of hyperbolic problems

Abstract We consider a class of numerical schemes for optimal control problems of hyperbolic conservation laws. We focus on finite-volume schemes using relaxation as a numerical approach to the optimality system. In particular, we study the arising numerical schemes for the adjoint equation and derive necessary conditions on the time integrator. We show that the resulting schemes are in particular asymptotic preserving for both, the adjoint and forward equation. We furthermore prove that higher-order time-integrator yields suitable Runge–Kutta schemes. The discussion includes the numerically interesting zero relaxation case.

A Mixed Finite Element Scheme for Optimal Control Problems with Pointwise State Constraints

In this paper, we propose a mixed variational scheme for optimal control problems with point-wise state constraints, the main idea is to reformulate the optimal control problems to a constrained minimization problem involving only the state, which is characterized by a fourth order variational inequality. Then mixed form based on this fourth order variational inequality is formulated and a direct numerical algorithm is proposed without the optimality conditions of underlying optimal control problems. The a priori and a posteriori error estimates are proved for the mixed finite element scheme. Numerical experiments confirm the efficiency of the new strategy.

Approximation of Rate-Independent Optimal Control Problems

This work introduces an abstract approximation scheme for optimal control problems in the energetic theory of rate-independent systems. This scheme builds upon the usual temporal semidiscretization and a $\Gamma$-convergence approximation of the functionals (in a full discretization, this approximation consists of a sequence of discrete functionals defined on finer and finer discrete spaces). The main new result of this work is a rigorous convergence analysis for this scheme. Interestingly, it turns out that the usual time discretization is incomplete as a basis for a fully discrete approximation of the optimal control problem, and a new nonlocal condition has to be introduced. Since this new condition is necessary and sufficient for a one-to-one correspondence of discrete-time and continuous-time solutions to rate-independent systems, we also gain a new theoretical insight, which might be of independent interest. At the heart of the derivation are so-called reverse approximation results, for which we provide extensions and refinements. To give a concrete example and to demonstrate how the abstract theory can be applied in a concrete problem, we present a full discretization with finite elements of the optimal control problem for a rate-independent partial differential inclusion.

An approach to controlled mechanical systems based on the multiobjective optimization technique

This paper discusses the multiobjective optimization techniques for a class of optimal control problems in mechanics. We deal with constrained nonlinear control systems described by the Euler-Lagrange or Hamilton equations and study the variational structure of the solution of the corresponding boundary-value problems. We also reduce the original ''mechanical'' problem to an auxiliary multiobjective optimization problem. This approach makes it possible to apply the effective theoretical and computational results from multiobjective programming to the original problem. We consider first order computational schemes for optimal control problems governed by mechanical systems and examine some illustrative examples.

Using dynamic programming with adaptive grid scheme for optimal control problems in economics

The study of the solutions of dynamic models with optimizing agents has often been limited by a lack of available analytical techniques to explicitly find the global solution paths. On the other hand, the application of numerical techniques such as dynamic programming to find the solution in interesting regions of the state was restricted by the use of fixed grid size techniques. Following Grüne (Numer. Math. 75 (3) (1997) 319; University of Bayreuth, submitted, 2003), in this paper an adaptive grid scheme is used for finding the global solutions of discrete time Hamilton–Jacobi–Bellman equations. Local error estimates are established and an adapting iteration for the discretization of the state space is developed. The advantage of the use of adaptive grid scheme is demonstrated by computing the solutions of one- and two-dimensional economic models which exhibit steep curvature, complicated dynamics due to multiple equilibria, thresholds (Skiba sets) separating domains of attraction and periodic solutions. We consider deterministic and stochastic model variants. The studied examples are from economic growth, investment theory, environmental and resource economics.

Approximation of integro-differential equations associated with piecewise deterministic process

Aim of this paper is to present an approximation scheme for optimal control problems of piecewise deterministic processes and corresponding integro-differential Hamilton–Jacobi–Bellman equations. The method is based on a discrete dynamic programming approach. We discretize the continuous process and the cost functional obtaining a discrete time optimal control problem. The corresponding dynamic programming equation gives an approximation of the integro-differential equation. The main feature of the method is the uniform convergence to the value function of the continuous control problem, which can be characterized as the unique weak solution (in viscosity sense) of the dynamic programming equation. Moreover, under appropriate assumptions, an error estimate on the truncation error is derived. It is worth noting that the method provides approximate feedback controls at any point of the grid without extra computations. An application of the approximation scheme to the numerical solution of an optimal control problem for a storage process is also detailed. © 1997 John Wiley & Sons, Ltd.