Relaxation of Optimal Control Problems Involving Time Dependent Subdifferential Operators

Solutions to optimal control problems obtained via direct methods are usually of low resolution. As the desired resolution becomes higher, the number of parameters in the resulting parameter optimization problem increases nonlinearly. A concatenated approach to trajectory optimization previously introduced by the authors to obtain high resolution optimal solutions with low storage requirements is applicable only to a specific class of problems, such as minimum time and minimum energy optimal control problems. The hodograph ballooning approach introduced in this paper generalizes the concatenated approach to the most general optimal control problem. Introduction The development of powerful nonlinear programming software [1, 2] has made direct approaches to trajectory optimization the method of choice for most practical applications [3-5]. The discretization of an optimal control problem to a finite dimensional parameter optimization problem avoids many of the troublesome pathologies that are typical for the optimization over infinite dimensional function spaces [6-8]. Furthermore, direct approaches usually do not. require guessing the switching structure, and the dynamical system is devoid of the nonintuitive costate variables. However, to achieve good resolution it is usually necessary to refine some type of discretization mesh, and, ultimately, to use a large number of design parameters. For typical gradient search methods, this increase in the dimension of the optimization problem has nearly always a negative effect on the convergence rate and the robustness. Additionally, CPU time and RAM requirements for each iteration increase with the second power of the number of parameters (third power for non-sparse solvers) so that direct optimization approaches can become quite expensive. In [9], a method was introduced that achieves excellent resolution without solving high-dimensional nonlinear programming problems. This was achieved by lo'Senior Engineer, Member AIAA. t Supervising Engineer, Member AIAA. 117 Research Drive, Hampton, VA 23666. Copyright ©1996 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved cally optimizing the trajectory on an iterative sequence of short, partly overlapping subarcs. However, this approach is not applicable to the most general problem formulation. Loosely sneaking, it was required in [9] that the cost state does not appear explicitly in the right-hand side of the equations of motion. The present paper introduces a method through which the approach introduced in [9] becomes applicable to the most general problem formulation. Application of this method, however, requires a priori knowledge of the optimal cost. In practice it is observed that the optimal cost can be obtained with very high precision with reasonably low order discretizations. Once a good guess for the optimal cost is obtained presented in this paper can be applied to refine the trajectory, or, if the optimal cost value is not known a priori with sufficient precision, it can be applied iteratively with the prescribed cost value adjusted in an outer loop. Problem Formulation Let us consider the following simple optimal control problem stated in Mayer form: ^ 4 ( x ( t f ) , t f ) (1 )

Iteration–discretization methods for variational inequalities over fixed point sets

We prove the existence of solutions of a differential variational inequality involving a prox-regular set in an infinite dimensional Hilbert space via a new existence result of a non-convex state-dependent sweeping process.

This article presents a unified approach to simultaneous solution of optimization and optimal control problems in batch distillation, operating under different modes of operation: variable, constant, or optimal reflux. The simplified, computationally efficient short‐cut method and a novel algorithm to solve the optimal control problems in batch distillation is the basis of this unified approach. The short‐cut method identifies the feasible region of operation essential for optimization and optimal control problems, and provides analytical partial derivatives of the model parameters crucial to the solution.The new algorithm for the solution of optimal combination of the maximum principle and NLP optimization techniques. It circumvents the problems associated with the maximum principle approach (iterative solution of a two‐point boundary value problem, undounded control variables, and inability to handle the simultaneous optimization and optimal control problem), and the coupled ODE discretization‐NLP optimization scheme for nonlinear models (higher system nonlinearities, multiplicity of solutions, sensitivity of convergence to initial guesses). This algorithm reduces the dimensionality of the problem, and the nature of the algorithm allows a common platform to optimal solutions of different operating conditions. This article also shows that different categories of the optimal control problems in batch distillation essentially involve the solution of the maximum distillate problem.

1 A Survey on Computational Optimal Control.- Issues in the Direct Transcription of Optimal Control Problems to Sparse Nonlinear Programs.- Optimization in Control of Robots.- Large-scale SQP Methods and their Application in Trajectory Optimization.- Solving Optimal Control and Pursuit-Evasion Game Problems of High Complexity.- 2 Theoretical Aspects of Optimal Control and Nonlinear Programming.- Continuation Methods In Boundary Value Problems.- Second Order Optimality Conditions for Singular Extremals.- Synthesis of Adaptive Optimal Controls for Linear Dynamic Systems.- Control Applications of Reduced SQP Methods.- Time Optimal Control of Mechanical Systems.- 3 Algorithms for Optimal Control Calculations.- Second Order Algorithm for Time Optimal Control of a Linear System.- An SQP-type Solution Method for Constrained Discrete-Time Optimal Control Problems.- Numerical Methods for Solving Differential Games, Prospective Applications to Technical Problems.- Construction of the Optimal Feedback Controller for Constrained Optimal Control Problems with Unknown Disturbances.- Repetitive Optimization for Predictive Control of Dynamic Systems under Uncertainty.- Optimal Control of Multistage Systems Described by High-Index Differential-Algebraic Equations.- A New Class of a High Order Interior Point Method for the Solution of Convex Semiinfinite Optimization Problems.- A Structured Interior Point SQP Method for Nonlinear Optimal Control Problems.- 4 Software for Optimal Control Calculations.- Automated Approach for Optimizing Dynamic Systems.- ANDECS: A Computation Environment for Control Applications of Optimization.- Application of Automatic Differentiation to Optimal Control Problems.- OCCAL: A mixed symbolic-numeric Optimal Control CALculator.- 5 Applications of Optimal Control.- A Robotic Satellite with Simplified Design.- Nonlinear Control under Constraints of a Biological System.- An Object-Oriented Approach to Optimally Describe and Specify a SCADA System Applied to a Power Network.- Near-Optimal Flight Trajectories Generated by Neural Networks.- Performance of a Feedback Method with Respect to Changes in the Air-Density during the Ascent of a Two-Stage-To-Orbit Vehicle.- Linear Optimal Control for Reentry Flight.- Steady-State Modelling of Turbine Engine with Controllers.- Shortest Paths for Satellite Mounted Robot Manipulators.- Optimal Control of the Industrial Robot Manutec r3.

In general, optimal control problems rely on numerically rather than analytically solving methods, due to their nonlinearities. The direct method, one of the numerically solving methods, is mainly to transform the optimal control problem into a nonlinear optimization problem with finite dimensions, via discretizing the objective functional and the forced dynamical equations directly. However, in the procedure of the direct method, the classical discretizations of the forced equations will reduce or affect the accuracy of the resulting optimization problem as well as the discrete optimal control. In view of this fact, more accurate and efficient numerical algorithms should be employed to approximate the forced dynamical equations. As verified, the discrete variational difference schemes for forced Birkhoffian systems exhibit excellent numerical behaviors in terms of high accuracy, long-time stability and precise energy prediction. Thus, the forced dynamical equations in optimal control problems, after being represented as forced Birkhoffian equations, can be discretized according to the discrete variational difference schemes for forced Birkhoffian systems. Compared with the method of employing traditional difference schemes to discretize the forced dynamical equations, this way yields faithful nonlinear optimization problems and consequently gives accurate and efficient discrete optimal control. Subsequently, in the paper we are to apply the proposed method of numerically solving optimal control problems to the rendezvous and docking problem of spacecrafts. First, we make a reasonable simplification, i.e., the rendezvous and docking process of two spacecrafts is reduced to the problem of optimally transferring the chaser spacecraft with a continuously acting force from one circular orbit around the Earth to another one. During this transfer, the goal is to minimize the control effort. Second, the dynamical equations of the chaser spacecraft are represented as the form of the forced Birkhoffian equation. Then in this case, the discrete variational difference scheme for forced Birkhoffian system can be employed to discretize the chaser spacecraft's equations of motion. With further discretizing the control effort and the boundary conditions, the resulting nonlinear optimization problem is obtained. Finally, the optimization problem is solved directly by the nonlinear programming method and then the discrete optimal control is achieved. The obtained optimal control is efficient enough to realize the rendezvous and docking process, even though it is only an approximation of the continuous one. Simulation results fully verify the efficiency of the proposed method for numerically solving optimal control problems, if the fact that the time step is chosen to be very large to limit the dimension of the optimization problem is noted.

We are concerned with a variational inclusion problem with two monotone mappings, and fixed-point problems of asymptotically nonexpansive mappings and pseudocontractive mappings. A viscosity method is introduced and studied for the two problems. A convergence theorem in norm of common solutions is established in the setting of infinite dimensional real Hilbert spaces.

In this paper, we propose a new modification of the Gradient Projection Algorithm and the Forward–Backward Algorithm. Using our proposed algorithms, we establish two strong convergence theorems for solving convex minimization problem, monotone variational inclusion problem and fixed point problem for demicontractive mappings in a real Hilbert space. Furthermore, we apply our results to solve split feasibility and optimal control problems. We also give two numerical examples of our algorithm in real Euclidean space of dimension 4 and in an infinite dimensional Hilbert space, to show the efficiency and advantage of our results.

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.

In this section we study a second class of evolution inclusions, which are driven by time-dependent subdifferential operators. The t-dependence of the convex function φ(t, ·) is an important feature of our inclusions and general enough to incorporate cases where domφ(t, ·) ⋂ domφ(s, ·) = O for t ≠ s. So the abstract framework of this chapter is suitable for the analysis of problems with time-varying obstacles. The structure of this chapter basically parallels that of chapter I.

Decomposition method for solving two optimal control problems and one optimization problem in batch fermentation is proposed. The problems are formulated based on a nonstructured mathematical model with slowly varying parameters and a finite cost criterion of maximum end production. Dependence of the model parameters on one physical or chemical parameter, which could easily be used as a control input is introduced analytically in the model equations and three model descriptions are obtained by nonlinear difference equations. Sensitivity functions of state trajectories towards slowly varying coefficients are introduced to account for model uncertainties. Based on them extended optimal control and optimization problems are formulated.

This paper focuses on addressing the linear quadratic (LQ) optimal control problem on an infinite time horizon for stochastic systems controlled by impulses. No constraint regarding the sign of the quadratic functional is applied. That is why our first concern is to find conditions which guarantee that the considered optimal control problem is well posed. Then, when the optimal control problem is well posed, it is natural to look for conditions which guarantee the attainability of the optimal control problem that is being evaluated. The main tool involved in the solution of the problems stated before is a backward jump matrix linear differential equation (BJMLDE) with a Riccati-type jumping operator. This is formulated using the matrix coefficients of the controlled system and the weight matrices of the performance criterion. We show that the well posedness of the optimal control problem under investigation is guaranteed by the existence of the maximal and bounded solution of the associated BJMLDE with a Riccati-type jumping operator. Further, we show that when the associated BJMLDE with a Riccati-type jumping operator has a maximal solution which satisfies a suitable sign condition, then the optimal control problem is attainable if and only if it has an optimal control in a state feedback form, or if and only if the maximal solution of the BJMLDE with a Riccati-type jumping operator is a stabilizing solution. In order to make the paper more self-contained, we present a set of conditions that correspond to the existence of the maximal solution of the BJMLDE satisfying the desired sign condition.

In this paper we investigate the use of optimal control techniques to improve Functional Electrical Stimulation (FES) for drop foot correction on hemiplegic patients. A model of the foot and the tibialis anterior muscle, the contraction of which is controlled by electrical stimulation has been established and is used in the optimal control problem. The novelty in this work is the use of the ankle accelerations and shank orientations (so-called external states) in the model, which have been measured on hemiplegic patients in a previous experiment using Inertial Measurement Units (IMUs). The optimal control problem minimizes the square of muscle excitations which serves the overall goal of reducing energy consumption in the muscle. In a first step, an offline optimal control problem is solved for test purposes and shows the efficiency of the FES optimal control for drop foot correction. In a second step, a Nonlinear Model Predictive Control (NMPC) problem - or online optimal control problem, is solved in a simulated environment. While the ulitmate goal is to use NMPC on the real system, i.e. directly on the patient, this test in simulation was meant to show the feasibility of NMPC for online drop foot correction. In the optimization problem, a set of fixed constraints of foot orientation was applied. Then, an original adaptive constraint taking into account the current ankle height, was introduced and tested. Comparisons between results under fixed and adaptive constraints highlight the advantage of the adaptive constraints in terms of energy consumption, where its quadratic sum of controls, obtained by NMPC, was three times lower than with the fixed constraint. This feasibility study was a first step in application of NMPC on real hemiplegic patients for online FES-based drop foot correction. The adaptive constraints method presents a new and efficient approach in terms of muscular energy consumption minimization.

This article addresses the optimal tracking control problem with prescribed performance for uncertain nonlinear systems subject to input constraint and unknown disturbances. First, a fixed‐time monotonic convergence function is introduced to restrain tracking error, and a nonlinear mapping technique is employed to transform the constrained error into an unconstrained variable, then the fixed‐time output tracking issue is boiled down to the boundedness problem of the transformed variable. With the aid of a nonquadratic cost function, the input constraint is encoded into the optimization problem. To solve the unknown disturbances, an auxiliary system and an auxiliary disturbance policy are constructed, and the optimal control problem is formulated as a two‐player zero‐sum game. Moreover, a Hamilton–Jacobi–Isaacs (HJI) equation associated with this nonquadratic zero‐sum game is established to give the optimal control and the worst‐case disturbance policy solution. Subsequently, to avoid using knowledge of the system dynamics, three neural network approximators, namely, actor, critic, and disturbance, which are tuned online and simultaneously for approximating the solution of HJI, are constructed based on the integral reinforcement learning algorithm. Theoretical analysis shows that the reconstructed error system states and the weight estimation errors are semi‐globally uniformly ultimately bounded. Finally, the simulation study further tests the availability of the proposed control strategy.

The main aim of this work is to introduce an implicit general iterative method for approximating a solution of a split variational inclusion problem with a hierarchical optimization problem constraint for a countable family of mappings, which are nonexpansive, in the setting of infinite dimensional Hilbert spaces. Convergence theorem of the sequences generated in our proposed implicit algorithm is obtained under some weak assumptions.