Multiple-interval pseudospectral approximation for nonlinear optimal control problems with time-varying delays

A multiple-interval pseudospectral scheme is developed for solving nonlinear optimal control problems with time-varying delays, which employs collocation at the shifted flipped Jacobi-Gauss–Radau points. The new pseudospectral scheme has the following distinctive features/abilities: (i) it can directly and flexibly solve nonlinear optimal control problems with time-varying delays without the tedious quasilinearization procedure and the uniform mesh restriction on time domain decomposition, and (ii) it provides a smart approach to compute the values of state delay efficiently and stably, and a unified framework for solving standard and delay optimal control problems. Numerical results on benchmark delay optimal control problems including challenging practical engineering problems demonstrate that the proposed pseudospectral scheme is highly accurate, efficient and flexible.     

On Generic Existence and Uniqueness in Nonconvex Optimal Control Problems

We derive conditions for generic existence and uniqueness of optimal control and trajectories for some class of finite-dimensional optimal control problems in the absence of traditional convexity assumptions. It is shown that for these problems existence and uniqueness of optimal control for a given initial point x is equivalent to the differentiability of optimal value functions at x. These results are obtained in the general framework of representation formulas for (sub-)gradients of inf-envelope functions which appear in nonsmooth analysis.     

Asymptotic properties of solutions of parametric optimal control problems with varying index of the singular arcs

We consider a family of parametric linear-quadratic optimal control problems with terminal and control constraints. This family has the specific feature that the class of optimal controls is changed for an arbitrarily small change in the parameter. In the perturbed problem, the behavior of the corresponding trajectory on noncritical arcs of the optimal control is described by solutions of singularly perturbed boundary value problems. For the solutions of these boundary value problems, we obtain an asymptotic expansion in powers of the small parameter ?. The asymptotic formula starts from a term of the order of 1/? and contains boundary layers. This formula is used to justify the asymptotic expansion of the optimal control for a perturbed problem in the family. We suggest a simple method for constructing approximate solutions of the perturbed optimal control problems without integrating singularly perturbed systems. The results of a numerical experiment are presented.     

Error estimates and superconvergence of semidiscrete mixed methods for optimal control problems governed by hyperbolic equations

In this paper, we investigate the L ^??(L ₂)-error estimates and superconvergence of the semidiscrete mixed finite elementmethods for quadratic optimal control problems governed by linear hyperbolic equations. The state and the co-state are discretized by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k(k ?? 0). We derive error estimates for approximation of both state and control. Moreover, we present the superconvergence analysis for mixed finite element approximation of the optimal control problems.     

Exterior penalty in optimal control problem with state-control constraints

We consider an abstract optimal control problem with additional equality and inequality state and control constraints, we use the exterior penalty function to transform the constrained optimal control problem into a sequence of unconstrained optimal control problems, under conditions in control lie in L ₁, the sequence of the solution to the unconstrained problem contains a subsequence converging of the solution of constrained problem, this convergence is strong when the problemis non convex, and is weak if the problemis convex in control. This generalizes the results of P.Nepomiastcthy [4] where he considered the control in the Hilbert space L ²(I,? ^m ).     

Parametric approach to optimal control

We consider the optimal control problem from view point of parametric aspects. We have examined two cases of the parameterized problems. First case describes the situation when the objective functional contains time t as a parameter. We also show how to apply the parametric optimization techniques, such as pathfollowing methods, for finding a nominal optimal control path.     

Generic well-posedness of nonconvex optimal control problems

The Tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions. In our previous work a generic existence and uniqueness result (with respect to variations of the integrand of the integral functional) without the convexity condition was established for a class of optimal control problems satisfying the Cesari growth condition. In this paper we extend this generic existence and uniqueness result to a class of optimal control problems in which the right-hand side of differential equations is also subject to variations.     

TACO: a toolkit for AMPL control optimization

We describe a set of extensions to the AMPL modeling language to conveniently model mixed-integer optimal control problems for ODE or DAE dynamic processes. These extensions are realized as AMPL user functions and suffixes and do not require intrusive changes to the AMPL language standard or implementation itself. We describe and provide TACO, a Toolkit for AMPL Control Optimization that reads AMPL stub.nl files and detects the structure of the optimal control problem. This toolkit is designed to facilitate the coupling of existing optimal control software packages to AMPL. We discuss requirements, capabilities, and the current implementation. Using the example of the multiple shooting code for optimal control MUSCOD-II, a direct and simultaneous method for DAE-constrained optimal control, we demonstrate how the problem information provided by the TACO toolkit is interfaced to the solver. In addition, we show how the MS-MINTOC algorithm for mixed-integer optimal control can be used to efficiently solve mixed-integer optimal control problems modeled in AMPL. We use the AMPL extensions to model three control problem examples and we discuss how those extensions affect the representation of optimal control problems. Solutions to these problems are obtained by using MUSCOD-II and MS-MINTOC inside the AMPL environment. A collection of further AMPL control models is provided on the web site http://mintoc.de. MUSCOD-II and MS-MINTOC have been made available on the NEOS Server for Optimization, using the TACO toolkit to enable input of AMPL models.     

Eigen series solutions to terminal-state tracking optimal control problems and exact controllability problems constrained by linear parabolic PDEs

Terminal-state tracking optimal control problems for linear parabolic equations are studied in this paper. The control objectives are to track a desired terminal state and the control is of the distributed type. Explicit solution formulae for the optimal control problems are derived in the form of eigen series. Pointwise-in-time L² norm estimates for the optimal solutions are obtained and approximate controllability results are established. Exact controllability is shown when the target state vanishes on the boundary of the spatial domain. One-dimensional computational results are presented which illustrate the terminal-state tracking properties for the solutions expressed by the series formulae.     

Recent advances in gradient algorithms for optimal control problems

This paper summarizes recent advances in the area of gradient algorithms for optimal control problems, with particular emphasis on the work performed by the staff of the Aero-Astronautics Group of Rice University. The following basic problem is considered: minimize a functionalI which depends on the statex(t), the controlu(t), and the parameter π. Here,I is a scalar,x ann-vector,u anm-vector, and π ap-vector. At the initial point, the state is prescribed. At the final point, the statex and the parameter π are required to satisfyq scalar relations. Along the interval of integration, the state, the control, and the parameter are required to satisfyn scalar differential equations. First, the sequential gradient-restoration algorithm and the combined gradient-restoration algorithm are presented. The descent properties of these algorithms are studied, and schemes to determine the optimum stepsize are discussed. Both of the above algorithms require the solution of a linear, two-point boundary-value problem at each iteration. Hence, a discussion of integration techniques is given. Next, a family of gradient-restoration algorithms is introduced. Not only does this family include the previous two algorithms as particular cases, but it allows one to generate several additional algorithms, namely, those with alternate restoration and optional restoration. Then, two modifications of the sequential gradient-restoration algorithm are presented in an effort to accelerate terminal convergence. In the first modification, the quadratic constraint imposed on the variations of the control is modified by the inclusion of a positive-definite weighting matrix (the matrix of the second derivatives of the Hamiltonian with respect to the control). The second modification is a conjugate-gradient extension of the sequential gradient-restoration algorithm. Next, the addition of a nondifferential constraint, to be satisfied everywhere along the interval of integration, is considered. In theory, this seems to be only a minor modification of the basic problem. In practice, the change is considerable in that it enlarges dramatically the number and variety of problems of optimal control which can be treated by gradient-restoration algorithms. Indeed, by suitable transformations, almost every known problem of optimal control theory can be brought into this scheme. This statement applies, for instance, to the following situations: (i) problems with control equality constraints, (ii) problems with state equality constraints, (iii) problems with equality constraints on the time rate of change of the state, (iv) problems with control inequality constraints, (v) problems with state inequality constraints, and (vi) problems with inequality constraints on the time rate of change of the state. Finally, the simultaneous presence of nondifferential constraints and multiple subarcs is considered. The possibility that the analytical form of the functions under consideration might change from one subarc to another is taken into account. The resulting formulation is particularly relevant to those problems of optimal control involving bounds on the control or the state or the time derivative of the state. For these problems, one might be unwilling to accept the simplistic view of a continuous extremal arc. Indeed, one might want to take the more realistic view of an extremal arc composed of several subarcs, some internal to the boundary being considered and some lying on the boundary. The paper ends with a section dealing with transformation techniques. This section illustrates several analytical devices by means of which a great number of problems of optimal control can be reduced to one of the formulations presented here. In particular, the following topics are treated: (i) time normalization, (ii) free initial state, (iii) bounded control, and (iv) bounded state.     

A semismooth Newton method for a class of semilinear optimal control problems with box and volume constraints

In this paper we consider optimal control problems subject to a semilinear elliptic state equation together with the control constraints 0≤u≤1 and ∫u=m. Optimality conditions for this problem are derived and reformulated as a nonlinear, nonsmooth equation which is solved using a semismooth Newton method. A regularization of the nonsmooth equation is necessary to obtain the superlinear convergence of the semismooth Newton method. We prove that the solutions of the regularized problems converge to a solution of the original problem and a path-following technique is used to ensure a constant decrease rate of the residual. We show that, in certain situations, the optimal controls take 0–1 values, which amounts to solving a topology optimization problem with volume constraint.     

Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems

We study the numerical approximation of distributed optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. Piecewise linear finite elements are used to approximate the control as well as the state. We prove that the L ²-error estimates are of order o(h), which is optimal according with the $C^{0,1}(\overline{\Omega})$ -regularity of the optimal control.     

Generic existence of Lipschitzian solutions of optimal control problems without convexity assumptions

The Tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions. In this paper we consider a large class of optimal control problems which is identified with a complete metric space of integrands without convexity assumptions and show that for a generic integrand the corresponding optimal control problem possesses a unique solution and this solution is Lipschitzian.     

Properties and numerical testing of a parallel global optimization algorithm

In the present paper, we have considered three methods with which to control the error in the homotopy analysis of elliptic differential equations and related boundary value problems, namely, control of residual errors, minimization of error functionals, and optimal homotopy selection through appropriate choice of auxiliary function H(x). After outlining the methods in general, we consider three applications. First, we apply the method of minimized residual error in order to determine optimal values of the convergence control parameter to obtain solutions exhibiting central symmetry for the Yamabe equation in three or more spatial dimensions. Secondly, we apply the method of minimizing error functionals in order to obtain optimal values of the convergnce control parameter for the homotopy analysis solutions to the Brinkman?CForchheimer equation. Finally, we carefully selected the auxiliary function H(x) in order to obtain an optimal homotopy solution for Liouville??s equation.     

A posteriori error estimates for hp finite element solutions of convex optimal control problems

In this paper, we present a posteriori error analysis for hp finite element approximation of convex optimal control problems. We derive a new quasi-interpolation operator of Clément type and a new quasi-interpolation operator of Scott-Zhang type that preserves homogeneous boundary condition. The Scott-Zhang type quasi-interpolation is suitable for an application in bounding the errors in L²-norm. Then hp a posteriori error estimators are obtained for the coupled state and control approximations. Such estimators can be used to construct reliable adaptive finite elements for the control problems.     

Convergence of distributed optimal control problems governed by elliptic variational inequalities

First, let u _g be the unique solution of an elliptic variational inequality with source term g. We establish, in the general case, the error estimate between $u_{3}(\mu)=\mu u_{g_{1}}+ (1-\mu)u_{g_{2}}$ and $u_{4}(\mu)=u_{\mu g_{1}+ (1-\mu) g_{2}}$ for ????[0,1]. Secondly, we consider a family of distributed optimal control problems governed by elliptic variational inequalities over the internal energy g for each positive heat transfer coefficient h given on a part of the boundary of the domain. For a given cost functional and using some monotony property between u ₃(??) and u ₄(??) given in Mignot (J.?Funct. Anal. 22:130?C185, 1976), we prove the strong convergence of the optimal controls and states associated to this family of distributed optimal control problems governed by elliptic variational inequalities to a limit Dirichlet distributed optimal control problem, governed also by an elliptic variational inequality, when the parameter h goes to infinity. We obtain this convergence without using the adjoint state problem (or the Mignot??s conical differentiability) which is a great advantage with respect to the proof given in Gariboldi and Tarzia (Appl. Math. Optim. 47:213?C230, 2003), for optimal control problems governed by elliptic variational equalities.     

Time optimal sampled-data controls for the heat equation

In this paper, we first design a time optimal control problem for the heat equation with sampled-data controls, and then use it to approximate a time optimal control problem for the heat equation with distributed controls.The study of such a time optimal sampled-data control problem is not easy, because it may have infinitely many optimal controls. We find connections among this problem, a minimal norm sampled-data control problem and a minimization problem, and obtain some properties on these problems. Based on these, we not only build up error estimates for optimal time and optimal controls between the time optimal sampled-data control problem and the time optimal distributed control problem, in terms of the sampling period, but we also prove that such estimates are optimal in some sense.     

Error estimates of mixed finite element methods for quadratic optimal control problems

In this paper, we investigate the error estimates for the solutions of optimal control problems by mixed finite element methods. The state and costate are approximated by Raviart-Thomas mixed finite element spaces of order k and the control is approximated by piecewise polynomials of order k. Under the special constraint set, we will show that the control variable can be smooth in the whole domain. We derive error estimates of optimal order both for the state variables and the control variable.     

L ∞-error estimates of triangular mixed finite element methods for optimal control problems governed by semilinear elliptic equations

We look at L ^∞-error estimates for convex quadratic optimal control problems governed by nonlinear elliptic partial differential equations. In so doing, use is made of mixed finite element methods. The state and costate are approximated by the lowest order Raviart-Thomas mixed finite element spaces, and the control, by piecewise constant functions. L ^∞-error estimates of optimal order are derived for a mixed finite element approximation of a semilinear elliptic optimal control problem. Finally, numerical tests are presented which confirm our theoretical results.     

A priori error estimates of finite volume element method for hyperbolic optimal control problems

In this paper,optimize-then-discretize,variational discretization and the finite volume method are applied to solve the distributed optimal control problems governed by a second order hyperbolic equation.A semi-discrete optimal system is obtained.We prove the existence and uniqueness of the solution to the semidiscrete optimal system and obtain the optimal order error estimates in L ∞(J;L 2)-and L ∞(J;H 1)-norm.Numerical experiments are presented to test these theoretical results.