Class Of Optimal Control Problems Research Articles

Fourier Finite Volume Element Method for Two Classes of Optimal Control Problems Governed by Elliptic PDEs on Complex Connected Domain

In this article, Fourier finite volume element method is provided to solve two classes of optimal control problems governed by elliptic PDEs. In two classes of problems, both distributed control and Dirichlet boundary control are given. We obtain the optimality systems of optimal control problems by the Lagrange multiplier method. Using the polar coordinates, the Fourier discretization is used in the azimuthal direction while the finite volume element approximation is used in the radial direction. In our proposed numerical method, the trial and test function spaces are carefully chosen to get accurate approximation. We use the Fourier finite volume method to approximate the control problems, the two-dimensional optimal control problem can be reduced to a one-dimensional problem. We not only observe that the convergent order of distributed control is second order, but also the convergent order of the Dirichlet boundary control is second order. Some numerical experiments are presented to illustrate accuracy and efficiency of our method.

Hamilton–Jacobi equations for optimal control on junctions with unbounded running cost functions

ABSTRACT This paper focuses on the viscosity solution approach to optimal control problems on junctions. Compared to Achdou et al. [Hamilton–Jacobi equations for optimal control on junctions and networks: ESAIM Control Optim. ESAIM Control Optim Calc Var. 2015;21(3):876–899] and Khang [Hamilton–Jacobi equations for optimal control on networks with entry or exit costs. ESAIM Control Optim Calc Var. 2018. (In press). EDP Sciences. <hal-01548133v2>], we work on a less restrictive set of assumptions. We show that the value function is a unique viscosity solution of an associated Hamilton–Jacobi equation, and present some further properties of it. In addition, the viscosity solution method is used to establish a necessary and sufficient condition for an optimal control in a class of optimal control problems.

Optimal Control Problems with Disorder

We consider a generalization of processes with disorder, namely processes with a vector disorder. For these problems, we consider a class of optimal control problems that do not detect the disorder. We propose a computational method for solving control problems on a finite time interval and with an objective functional defined at the end of the interval, based on the use of the martingale technique. We consider a computational experiment for a model with two barriers and two stopping times.

On a Class of Optimal Control Problems for Functional Differential Systems

A linear functional differential control system of general form with aftereffect is considered. An optimal control problem with linear constraints on the state and control variables is studied. The control is realized by a linear operator of general form. The cases of distributed and lumped delay in the control loop, as well as the case of impulsive control, are covered. The Cauchy matrix is used to reduce the problem under consideration to a problem formulated only in terms of control variables with the use of some auxiliary variables linked with the defining relations for the Cauchy matrix of the system. In the case where the control is chosen from a finite-dimensional subspace of the control space, a problem effectively solvable by standard software tools is written explicitly. An example of an applied optimal control problem that arises in economic dynamics is presented. A class of hybrid systems (systems with continuous and discrete times) reducible to the system under consideration is described.

Uniqueness of $$\beta $$-Viscosity Solutions of Hamilton–Jacobi Equations and Applications to a Class of Optimal Control Problems

This paper is concerned with an infinite horizon optimal control with unbounded value functions for a class of Hamilton–Jacobi equations (HJEs) in Banach spaces. Based on the uniqueness of \(\beta \)-viscosity solutions of HJEs, which is first established in this paper, necessary and sufficient optimality conditions are derived.

Numerical Solution for Classical Optimal Control Problem Governing by Hyperbolic Partial Differential Equation via Galerkin Finite Element-Implicit method with Gradient Projection Method

This paper deals with the numerical solution of the discrete classical optimal control problem (DCOCP) governing by linear hyperbolic boundary value problem (LHBVP). The method which is used here consists of: the GFEIM " the Galerkin finite element method in space variable with the implicit finite difference method in time variable" to find the solution of the discrete state equation (DSE) and the solution of its corresponding discrete adjoint equation, where a discrete classical control (DCC) is given. The gradient projection method with either the Armijo method (GPARM) or with the optimal method (GPOSM) is used to solve the minimization problem which is obtained from the necessary condition for optimality of the DCOCP to find the DCC.An algorithm is given and a computer program is coded using the above methods to find the numerical solution of the DCOCP with step length of space variable , and step length of time variable . Illustration examples are given to explain the efficiency of these methods. The results show the methods which are used here are better than those obtained when we used the Gradient method (GM) or Frank Wolfe method (FWM) with Armijo step search method to solve the minimization problem.

Optimal control problems with control complementarity constraints: existence results, optimality conditions, and a penalty method

ABSTRACTA special class of optimal control problems with complementarity constraints on the control functions is studied. It is shown that such problems possess optimal solutions whenever the underlying control space is a first-order Sobolev space. After deriving necessary optimality conditions of strong stationarity-type, a penalty method based on the Fischer–Burmeister function is suggested and its theoretical properties are analyzed. Finally, the numerical treatment of the problem is discussed and results of computational experiments are presented.

Parameter space optimization towards integrated mechatronic design for uncertain systems with generalized feedback constraints

There is an increasing trend to apply integrated mechatronic design approaches in precision engineering to synthesize key mechanical and controller parameters simultaneously. However, such technique is yet to be mature, due to the constraints imported by mechanical design and feedback control, as well as the existence of model uncertainties. In this work, we treat the integrated mechatronic design problem as a controller optimization problem with structural constraints. We start from the case when the composite feedback gain matrix (CFGM) has some elements either being zero or with equal or opposite relationships. First, algorithms are proposed to factorize the CFGM. Secondly, the feedback constraints are transformed from the state space to an extended parameter space. In this way, the design problem is reformulated as minimizing the ℋ2-norm upper bound of the closed-loop system transmittance from the exogenous disturbance to the regulated variables over the intersection of convex and non-convex domains. Eventually, cutting-plane-based numerical procedures are developed to obtain a global optimal solution, and the closed-loop robust stability is ensured with guaranteed performance. An illustrative example on a flexure-linked biaxial gantry stage is presented to reveal the practical appeal of the proposed approach. This approach is extensively applicable to a class of optimal control problems, such as controller synthesis problem with prescribed sparsity pattern, decentralized control problem with/without structural constraints, etc.

On an Optimal Control Problem with Discontinuous Integrand

We consider an optimal control problem for an autonomous differential inclusion with free terminal time and a mixed functional which contains the characteristic function of a given open set M ⊂ ℝn in the integral term. The statement of the problem weakens the statement of the classical optimal control problem with state constraints to the case where the presence of admissible trajectories of the system in the set M is physically allowed but undesirable due to safety or instability reasons. Using an approximation approach, necessary conditions for the optimality of an admissible trajectory are obtained in the form of Clarke’s Hamiltonian inclusion. The result involves a nonstandard stationarity condition for the Hamiltonian. As in the case of the problem with a state constraint, the obtained necessary optimality conditions may degenerate. Conditions guaranteeing their nondegeneracy and pointwise nontriviality are presented. The results obtained extend the author’s previous results to the case of a problem with free terminal time and more general functional.

Necessary optimality conditions for average cost minimization problems

Control systems involving unknown parameters appear a natural framework for applications in which the model design has to take into account various uncertainties. In these circumstances the performance criterion can be given in terms of an average cost, providing a paradigm which differs from the more traditional minimax or robust optimization criteria. In this paper, we provide necessary optimality conditions for a nonrestrictive class of optimal control problems in which unknown parameters intervene in the dynamics, the cost function and the right end-point constraint. An important feature of our results is that we allow the unknown parameters belonging to a mere complete separable metric space (not necessarily compact).

一类无穷时域上布尔网络最优控制问题的求解方法

一类无穷时域上布尔网络最优控制问题的求解方法

Optimal control of linear PDEs using occupation measures and SDP relaxations

This paper addresses the problem of solving a class of optimal control problems (OCPs) with infinite-dimensional linear state constraints involving Riesz-spectral operators. Each instance within this class has time/control-dependent polynomial Lagrangian cost and control constraints described by polynomials. We first perform a state-mode discretization of the Riesz-spectral operator. Then we approximate the resulting finite-dimensional OCPs by using a previously known hierarchy of semidefinite relaxations. Under certain compactness assumptions, we provide a converging hierarchy of semidefinite programming relaxations whose optimal values yield lower bounds for the initial OCP. We illustrate our method by two numerical examples, involving a diffusion partial differential equation and a wave equation. We also report on the related experiments.

Application of the Variational Method for Solving Inverse Problems of Optimal Control

For optimal control problems, a new approach based on the search for an extremum of a special functional is proposed. The differential problem is reformulated as an ill-posed variational inverse problem. Taking into account ill-posedness leads to a stable numerical minimization procedure. The method developed has a high degree of generality, since it allows one to find special controls. Several examples of interest concerning the solution of classical optimal control problems are considered.

A measure theoretic approach to traffic flow optimisation on networks

We consider a class of optimal control problems for measure-valued nonlinear transport equations describing traffic flow problems on networks. The objective is to minimise/maximise macroscopic quantities, such as traffic volume or average speed, controlling few agents, e.g. smart traffic lights and automated cars. The measure theoretic approach allows to study in a same setting local and non-local drivers interactions and to consider the control variables as additional measures interacting with the drivers distribution. We also propose a gradient descent adjoint-based optimisation method, obtained by deriving first-order optimality conditions for the control problem, and we provide some numerical experiments in the case of smart traffic lights for a 2–1 junction.

Stable Wireless Network Control Under Service Constraints

We consider the design of wireless queueing network control policies with particular focus on combining stability with additional application-dependent requirements. Thereby, we consequently pursue a cost function based approach that provides the flexibility to incorporate constraints and requirements of particular services or applications. As typical examples of such requirements, we consider the reduction of buffer underflows in case of streaming traffic, and energy efficiency in networks of battery powered nodes. Compared with the classical throughput optimal control problem, such requirements significantly complicate the control problem. We provide easily verifiable theoretical conditions for stability, and, additionally, compare various candidate cost functions applied to wireless networks with streaming media traffic. Moreover, we demonstrate how the framework can be applied to the problem of energy efficient routing, and we demonstrate the application of our framework in cross-layer control problems for wireless multihop networks, using an advanced power control scheme for interference mitigation, based on successive convex approximation. In all scenarios, the performance of our control framework is evaluated using extensive numerical simulations.

A three-step decomposition method for solving the minimum-fuel geostationary station keeping of satellites equipped with electric propulsion

In this paper, a control scheme is elaborated in order to perform the station keeping of a geostationary satellite equipped with electric propulsion while minimizing the fuel consumption. The use of electric thrusters imposes to take into account some additional non linear and operational constraints that make the overall station keeping optimal control problem difficult to solve directly. That is why the station keeping problem is decomposed in three successive control problems. The first one consists in solving a classical optimal control problem with an indirect method initialized by a direct method without enforcing the thrusters operational constraints. Starting from this non feasible solution for the genuine problem, the thrusters operating constraints are incorporated in the second problem, whose solution produces a feasible but non optimal control profile via two different ways. Finally, the third optimizes the commutation times thanks to a method borrowed to the switched systems theory. Simulation results on a realistic example validate the benefit of this particular control scheme in the reduction of the fuel consumption for the geostationary station keeping problem.

Optimal control of a nonconvex perturbed sweeping process

The paper concerns optimal control of discontinuous differential inclusions of the normal cone type governed by a generalized version of the Moreau sweeping process with control functions acting in both nonconvex moving sets and additive perturbations. This is a new class of optimal control problems in comparison with previously considered counterparts where the controlled sweeping sets are described by convex polyhedra. Besides a theoretical interest, a major motivation for our study of such challenging optimal control problems with intrinsic state constraints comes from the application to the crowd motion model in a practically adequate planar setting with nonconvex but prox-regular sweeping sets. Based on a constructive discrete approximation approach and advanced tools of first-order and second-order variational analysis and generalized differentiation, we establish the strong convergence of discrete optimal solutions and derive a complete set of necessary optimality conditions for discrete-time and continuous-time sweeping control systems that are expressed entirely via the problem data.

Free-endpoint optimal control of inhomogeneous bilinear ensemble systems

Optimal control of bilinear systems has been a well-studied subject in the areas of mathematical and computational optimal control. However, effective methods for solving emerging optimal control problems involving an ensemble of deterministic or stochastic bilinear systems are underdeveloped. These burgeoning problems arise in diverse applications from quantum control and molecular imaging to neuroscience. In this work, we develop an iterative method to find optimal controls for an inhomogeneous bilinear ensemble system with free-endpoint conditions. The central idea is to represent the bilinear ensemble system at each iteration as a time-varying linear ensemble system, and then solve it in an iterative manner. We analyze convergence of the iterative procedure and discuss optimality of the convergent solutions. The method is directly applicable to solve the same class of optimal control problems involving a stochastic bilinear ensemble system driven by independent additive noise processes. We demonstrate the robustness and applicability of the developed iterative method through practical control designs in neuroscience and quantum control.

Schur complement preconditioners for multiple saddle point problems of block tridiagonal form with application to optimization problems

Abstract The importance of Schur-complement-based preconditioners is well established for classical saddle point problems in $\mathbb{R}^N \times \mathbb{R}^M$. In this paper we extend these results to multiple saddle point problems in Hilbert spaces $X_1\times X_2 \times \cdots \times X_n$. For such problems with a block tridiagonal Hessian and a well-defined sequence of associated Schur complements, sharp bounds for the condition number of the problem are derived, which do not depend on the involved operators. These bounds can be expressed in terms of the roots of the difference of two Chebyshev polynomials of the second kind. If applied to specific classes of optimal control problems the abstract analysis leads to new existence results as well as to the construction of efficient preconditioners for the associated discretized optimality systems.

Stabilization of strictly dissipative discrete time systems with discounted optimal control

We consider stabilization of an equilibrium point via infinite horizon discounted optimal control in discrete-time. In addition to applications in economics and social sciences, discounted optimal control is a commonly used numerical technique guaranteeing solvability of certain classes of optimal control problems. In this paper, we present conditions based on strict dissipativity that ensure that the optimally controlled system is asymptotically stable or practically asymptotically stable. These conditions are shown to be complementary to recently proposed conditions based on a detectability property. Illustrative examples are provided.