Kind Of Optimal Control Problem Research Articles

On optimal control in a nonlinear interface problem described by hemivariational inequalities

Abstract This article is devoted to the existence of optimal controls in various control problems associated with a novel nonlinear interface problem on an unbounded domain with non-monotone set-valued transmission conditions. This interface problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models non-monotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality (HVI), which, however, lives on the unbounded domain, and thus cannot be analyzed in a reflexive Banach space setting. Boundary integral methods lead to another HVI that is amenable to functional analytic methods using standard Sobolev spaces on the interior domain and Sobolev spaces of fractional order on the coupling boundary. Broadening the scope of this article, we consider extended real-valued HVIs augmented by convex extended real-valued functions. Under a smallness hypothesis, we provide existence and uniqueness results; moreover, we establish a stability result with respect to the extended real-valued function as a parameter. Based on the latter stability result, we prove the existence of optimal controls for four kinds of optimal control problems: distributed control on the bounded domain, boundary control, simultaneous boundary-distributed control governed by the interface problem, and control of the obstacle driven by a related bilateral obstacle interface problem.

Error estimates for the optimal control of Navier-Stokes equations using curvature based stabilization

This paper presents a family of implicit-explicit (IMEX) time stepping scheme for the optimal control problem of the unsteady Navier-Stokes equations (NSE). The main feature of this kind of optimal control problem is that stabilization terms are proportional to discrete curvature of the solutions. First and second order optimality conditions are used for optimality of the curvature based stabilized Navier-Stokes equations. Complete stability and error analyses of state, adjoint and control variables are presented. Numerical experiments verify theoretical findings and illustrate the improvement of approximate solutions enhancing the efficiency of numerical scheme.

Equivalence of three kinds of optimal control problems for linear heat equations with memory

This paper studies an equivalence theorem for three different kinds of optimal control problems, which are optimal time control problems, optimal norm control problems, and optimal target control problems. The controlled systems in this paper are internally controlled linear heat equations with memory.

Optimal control and stabilization for linear continuous‐time mean‐field systems with delay

This paper studies optimal control and stabilization problems for continuous-time mean-field systems with input delay, which are the fundamental development of control and stabilization problems for mean-field systems. There are two main contributions: (1) To the best of the authors' knowledge, the present paper is the first to establish the necessary and sufficient solvability condition for this kind of optimal control problem with input delay, and to derive the analytical form of an optimal controller through overcoming the obstacle that separation principle no longer holds for multiplicative-noise systems. (2) For the stabilization problem, under the assumption of exact observability, it is strictly proven that the system is stabilizable if and only if the algebraic Riccati equation has a unique positive definite solution.

Optimal control for uncertain random singular systems with multiple time-delays

Chance theory provides us a useful tool to solve an optimal control problem with indeterminacy composing of both uncertainty and randomness. Based on chance theory, this paper studies an optimal control for uncertain random singular systems with multiple time-delays. First, an uncertain random singular system with multiple time-delays is introduced, and then the corresponding optimal control problem is established. The equivalent relationship between this problem and the optimal control problem for standard uncertain random systems is derived. Then the appropriate recurrence equations are proposed according to the dynamic programming method. Furthermore, two kinds of optimal control problems are discussed. The optimal control inputs and respective optimal values of the problems are provided via the solvability of the obtained equations. Finally, a numerical example is presented to show the effectiveness of our theoretical results.

Stochastic maximum principle for optimal control problem with a stopping time cost functional

In this study, we consider an optimal control problem driven by a stochastic differential system with a stopping time terminal cost functional. To solve the optimal control problem under the stopping time terminal cost functional, we introduce a multi-time state optimal control systems, and prove that the multi-time state optimal control systems is a near-optimal control problem. We use the near-optimal control problem to take place the original optimal control problem under the cost functional with a stopping time terminal. Then, we establish the stochastic maximum principle for this kind of optimal control problem by introducing a discrete terminal system. Finally, we provide an example to describe the main results of this study.

A stochastic maximum principle for partially observed stochastic control systems with delay

This paper deals with partially-observed optimal control problems for the state governed by stochastic differential equation with delay. We develop a stochastic maximum principle for this kind of optimal control problems using a variational method and a filtering technique. Also, we establish a sufficient condition without assumption of the concavity. Two examples that shed light on the theoretical results are established in the paper. In particular, in the example of an optimal investment problem with delay, its numerical simulation shows the effect of delay via a discretization technique for forward–backward stochastic differential equations (FBSDEs) with delay and anticipate terms.

Optimal control of conditioned processes with feedback controls

We consider a class of closed loop stochastic optimal control problems in finite time horizon, in which the cost is an expectation conditional on the event that the process has not exited a given bounded domain. An important difficulty is that the probability of the event that conditionates the strategy decays as time grows. The optimality conditions consist of a system of partial differential equations, including a Hamilton-Jacobi-Bellman equation (backward w.r.t. time) and a (forward w.r.t. time) Fokker-Planck equation for the law of the conditioned process. The two equations are supplemented with Dirichlet conditions. Next, we discuss the asymptotic behavior as the time horizon tends to +∞. This leads to a new kind of optimal control problem driven by an eigenvalue problem related to a continuity equation with Dirichlet conditions on the boundary. We prove existence for the latter. We also propose numerical methods and supplement the various theoretical aspects with simulations.

A Necessary Condition for Optimal Control of Forward-Backward Stochastic Control System with Lévy Process in Nonconvex Control Domain Case

This paper analyzes one kind of optimal control problem which is described by forward-backward stochastic differential equations with Lévy process (FBSDEL). We derive a necessary condition for the existence of the optimal control by means of spike variational technique, while the control domain is not necessarily convex. Simultaneously, we also get the maximum principle for this control system when there are some initial and terminal state constraints. Finally, a financial example is discussed to illustrate the application of our result.

Optimal control of backward doubly stochastic system

An optimal control problem for backward doubly stochastic system is considered, where the control domain is not necessarily convex. By the method of classical spike variation and duality technique, one necessary condition and one sufficient condition are established for this kind of optimal control problem. The related adjoint process is characterised by the solution of a forward doubly stochastic differential equation, which formulates a forward–backward doubly stochastic differential equation coupled with the state equation. As an illustration, the authors' theoretical results are applied to study an optimal harvesting problem and a linear-quadratic optimal control problem. Moreover, the corresponding maximum principle with an initial state constraint is obtained by Ekeland's variational principle.

A Minimum-Fuel Fixed-Time Low-Thrust Rendezvous Solved with the Switching Systems Theory

In this paper, a fuel optimal rendezvous problem is tackled in the Hill-Clohessy-Wiltshire framework with several operational constraints as bounds on the thrust, non linear non convex and disjunctive operational constraints (on-off profile of the thrusters, minimum elapsed time between two consecutive firings...). An indirect method and a decomposition technique have already been combined in order to solve this kind of optimal control problem with such constraints. Due to a great number of parameters to tune, satisfactory results are hard to obtain and are sensitive to the initial condition. Assuming that no singular arc exists, it can be shown that the optimal control exhibits a bang-bang structure whose optimal switching times are to be found. Noticing that a system with a bang-bang control profile can be considered as two subsystems switching from one with control on to with control off, and vice-versa, a technique coming from the switching systems theory is used in order to optimise the switching times.

A novel approach for the generation of complex humanoid walking sequences based on a combination of optimal control and learning of movement primitives

We combine optimal control and movement primitive learning in a novel way for the fast generation of humanoid walking movements and demonstrate our approach at the example of the humanoid robot HRP-2 with 36 degrees of freedom. The present framework allows for an efficient computation of long walking sequences consisting of feasible steps of different kind: starting steps from a static posture, cyclic steps or steps with varying step lengths, and stopping motions back to a static posture. Together with appropriate sensors and high level decision strategies this approach provides an excellent basis for an adaptive walking generation on challenging terrain. Our framework comprises a movement primitive model learned from a small number of example steps that are dynamically feasible and minimize an integral mean of squared torques. These training steps are computed by solving three different kinds of optimal control problems that are restricted by the whole-body dynamics of the robot and the gait cycle. The movement primitive model decomposes the joint angles, pelvis orientation and ZMP trajectories in the example data into a small number of primitives, which effectively deals with the redundancy inherent in highly articulated motion. New steps can be composed by weighted combinations of these primitives. The mappings from step parameters to weights are learned with a Gaussian process approach, the contiguity of subsequent steps is promoted by conditioning the beginning of a new step on the end of the current one. Each step can be generated in less than a second, because the expensive optimal control computations, which take several hours per step, are shifted to the precomputational off-line phase. We validate our approach in the virtual robot simulation environment OpenHRP and study the effects of different kernels and different numbers of primitives. We show that the robot can execute long walking sequences with varying step lengths without falling, and hence that feasibility is transferred from optimized to generated motions. Furthermore, we demonstrate that the generated motions are close to torque optimality on the interior parts of the steps but have higher torques than their optimized counterparts on the steps boundaries. Having passed the validation in the robot simulator, we plan to tackle the transfer of this approach to the real platform HRP-2 as a next step.

Equivalence of three different kinds of optimal control problems for Stokes equations

This article presents an equivalence theorem for three different kinds of optimal control problems, which are optimal target control problems, optimal norm control problems, and optimal time control problems. Controlled systems in this study are internally controlled Stokes equations.

Necessary optimality conditions for optimal distributed and (Neumann) boundary control of Burgers equation

In this paper, we investigate the optimal control of the Burgers equation. For both optimal distributed and (Neumann) boundary control problems, the Dubovitskii and Milyutin functional analytical approach is adopted in investigation of the Pontryagin maximum principles of the systems. The necessary optimality conditions are, respectively, presented for two kinds of optimal control problems in both fixed and free final horizon cases, four extremum problems in all. Moreover, in free final horizon case, the assumptions of admissible control set on convexity and non-empty interior are removed so that it can be any set including an interesting case contains only finite many points. Finally, a remark on how to utilize the obtained results is also made for the illustration.

Two Equivalence Theorems of Different Kinds of Optimal Control Problems for Schrödinger Equations

This paper builds up two equivalence theorems for different kinds of optimal control problems of internally controlled Schrodinger equations. The first one is an equivalence of the minimal norm and the minimal time control problems where the target sets are the origin of the state space. (The minimal time control problems are also called the first type of optimal time control problems.) The second one is an equivalence of three optimal control problems: optimal target control problems, optimal norm control problems, and the second type of optimal time control problems.

Finite element methods for elliptic optimal control problems with boundary observations

We study in this paper the finite element approximations to elliptic optimal control problems with boundary observations. The main feature of this kind of optimal control problems is that the observations or measurements are the outward normal derivatives of the state variable on the boundary, this reduces the regularity of solutions to the optimal control problems. We propose two kinds of finite element methods: the standard FEM and the mixed FEM, to efficiently approximate the underlying optimal control problems. For both cases we derive a priori error estimates for problems posed on polygonal domains. Some numerical experiments are carried out at the end of the paper to support our theoretical findings.

Optimal control of beer fermentation processes with Lipschitz-constraint on the control

Nowadays, one can find in almost all industrial products a trail of mathematical optimization. In particular, the theory and algorithms of optimal control have helped in various fields to reduce the production time and to improve the quality of the considered products. As a special class of applications, two optimal control formulations for the fermentation process of beer are presented. The reactions of the fermentation processes are modelled by a system of ordinary differential equations that are steered by the heating and cooling of the involved substrates. The occurring physical limitations, such as temperature limits and the requirement to avoid scenarios with unrealistic bang-bang controls, give rise to optimal control problems with general control constraints. Therefore, this paper has reviewed a forward–backward sweeping method for solving these kinds of optimal control problems and presents encouraging numerical results that were obtained by this approach. Copyright © 2014 The Institute of Brewing & Distilling

Two kinds of optimal controls for probabilistic mix-valued logical dynamic networks

This paper addresses two kinds of optimal control problems of probabilistic mix-valued logical control networks by using the semi-tensor product of matrices, and presents a number of new results on the optimal finite-horizon control and the first-passage model based control problems, respectively. Firstly, the probabilistic mix-valued logical control network is expressed in an algebraic form by the semi-tensor product method, based on which the optimal finite-horizon control problem is studied and a new algorithm for choosing a sequence of control actions is established to minimize a given cost functional over finite steps. Secondly, the first-passage model of probabilistic mix-valued logical networks is given and a new algorithm for designing the optimal control scheme is proposed to maximize the corresponding probability criterion. Finally, an illustrative example is studied to support our new results/algorithms.

On the optimal harvesting of size-structured population dynamics

This work is concerned with a kind of optimal control problem for a size-structured biological population model. Well-posedness of the state system and an adjoint system are proved by means of Banach’s fixed point theorem. Existence and uniqueness of optimal control are shown by functional analytical approach. Optimality conditions describing the optimal strategy are established via tangent and normal cones technique. The results are of the first ones for this novel structure.

Equivalence of Three Different Kinds of Optimal Control Problems for Heat Equations and Its Applications

This paper presents an equivalence theorem for three different kinds of optimal control problems, which are optimal target control problems, optimal time control problems, and optimal norm control problems. Controlled systems in this study are internally controlled heat equations. With the aid of this theorem, we establish an optimal norm feedback law and build up some explicit algorithms for solutions of optimal norm and optimal time control problems.