We develop the method of discrete approximations, which allows us to adequately replace the original optimal control problem by a sequence of wellposed finitedimensional optimization problems whose optimal solutions strongly converge to that of the controlled perturbed sweeping process. The main goal of this paper is developing the method of discrete approximations to derive necessary optimality conditions for a class of constrained sweeping processes with nonsmooth perturbations. Obtaining finite difference approximations using function values at equally spaced sample points is an important problem in numerical analysis. Keywords sequential quadratic programming siam journal discrete approximation coercivity condition euler approximation. The term w t w to will be normally distributed with mean zero and variance t t 0. Error analysis of discrete approximations to bangbang. Optimal control of a perturbed sweeping process via. The main contribution of this paper is that the optimal control problems with. Discrete time optimal control problems, constrained optimization problem, parallel band solver, multiprocessor implementation, shared memory multiprocessor. An important class of continuoustime optimal control problems are the socalled linearquadratic optimal control problems where the objective functional j in 3. Successive approximation approach of optimal control for. Discrete approximations to optimal trajectories using direct. Proceedings of the 1999 ieee international symposium on computer aided control system design cat.
Discrete approximations of the hamiltonjacobi equation for an optimal control problem of a di erentialalgebraic system. The recent work of jingqing han sheds lights on this problem and is introduced. Discrete approximations of continuous distributions by maximum entropy economics letters, vol. Linea rquadratic regulato r discrete optimal control discrete hamiltonjacobi eq. Discretetime linear systems discretetime linear systems discretetime linear system 8 pdf 75 kb.
Insection 3, we formulate the optimal dual control prob. The constrained state estimation problem can be reformulated as a series of optimal control problems cf. Abstract pdf 3753 kb 1995 consistency of primaldual approximations for convex optimal control problems. Discrete approximations to continuous optimal control.
Legendre pseudospectral approximations of optimal control. In section 2 we study convergence properties of the optimal value and optimal solutions. In this way, one of our main results is related to the order of approximationof the adjoint system of the discrete optimal control problem to that of the continuous one. In particular, a time optimal control law is constructed in.
Discrete time optimal control applied to pest control problems. Singularperturbation method for discrete models of. Discrete approximations to continuous optimal control problems. The closedloop and openloop optimal controls of a singularly perturbed continuous system are considered by means of their discrete models. Optimal control theory is a mature mathematical discipline with numerous applications. Frederic bonnans, philippe chartier, hasnaa zidani to cite this version. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to. Finiteapproximationerrorbased optimal control approach. Optimal recursive estimation, kalman lter, zakai equation.
In particular, we introduce the discrete time method of successive approximations msa, which is. Discrete approximations to optimal trajectories using. Optimization online optimal control of differential inclusions. Approximate maximum principle for discrete approximations of optimal control systems with nonsmooth objectives and endpoint constraints. This problem can be reduced to dynamic optimization of a stateconstrained unbounded differential inclusion with highly irregular data that cannot be treated. Optimization online optimal control of differential. Analysis of finite difference approximations of an optimal. Lecture slides dynamic programming and stochastic control. Discretetime optimal control problems, constrained optimization problem, parallel band solver, multiprocessor implementation, shared memory multiprocessor. Using this continuoustime relationship between the differential and integral costate, it is shown that the discrete approximations of the differential costate using lg and lgr collocation are related to the corresponding discrete approximations. T and y 2 rn is said to be the optimal value function.
Pdf the paper addresses an optimal control problem for a perturbed sweeping process of the rateindependent hysteresis type described by a controlled. Mordukhovich minsk received october 29,1976 the approximation of continuoustime optimal control problems by sequences of finitedimensional discrete time optimization problems, arising from difference replacement of derivatives, is investigated. On difference approximations of optimal control systems. Solution of discretetime optimal control problems on. The algorithm presented here solved the approximation problem for an arbitrary linear functional. Discrete approximations of the hamiltonjacobi equation for an optimal control problem of a di erentialalgebraic system j. Such a discretetime control system consists of four major parts.
Convergence of discretetime approximations of constrained. This allows one to characterize necessary conditions for optimality and develop training algorithms that do not rely on gradients with respect to the trainable parameters. In fact, as optimal control solutions are now often implemented digitally, contemporary control theory is now primarily concerned with discrete time systems and solutions. An optimal control approach to deep learning and applications to discreteweight neural networks. View enhanced pdf access article on wiley online library html view download pdf for offline viewing. Duality of optimal control and optimal estimation including new results. This is again a discrete analogue of the wellknown result that the h j equation applied to linear hamiltonian systems reduces to the riccati equation see, e. Optimal control theory is a branch of applied mathematics that deals with finding a control law for a dynamical system over a period of time such that an objective function is optimized.
Discrete approximations of the hamiltonjacobi equation. The difference between the two is that, in optimal control theory, the optimizer is a function, not just a single value. The latter framework mostly concerns a new class of optimal control problems. Legendre pseudospectral approximations of optimal control problems 3. A familiar example is simpsons rule for numerical integration. A survey of some of the earlier work is given by polak in 34. Numerical methods for solving optimal control problems. Discrete approximations to optimal trajectories using direct transcription and nonlinear programming. Discrete approximations and optimal control of nonsmooth. Solving the optimal control problems, however, is computationally demanding, because the problem dimension grows. Optimal control problems for sweeping processes have been recently recognized among the most interesting and challenging.
Approximate maximum principle for discrete approximations. Hence, by drawing replicas of this random variable, we can obtain exact replicas for s t at any t, t 0 structure exploitation, calculation of gradients matthias gerdts indirect, direct, and function space methods optimal control problem indirect method ibased on necessary optimality conditions minimum principle i leads to a boundary value problem bvp i bvp needs to be. This allows one to characterize necessary conditions for optimality and develop training algorithms that do not rely on gra. Using this continuoustime relationship between the differential and integral costate, it is shown that the discrete approximations of the differential costate using legendregauss and legendregaussradau collocation are related to the corresponding discrete approximations of the integral costate. Discrete hamiltonjacobi theory and discrete optimal control. Discrete time optimal control applied to pest control problems 481 the paper is organized as follows. The paper is devoted to the study of a new class of optimal control problems. The latter framework mostly concerns a new class of optimal control problems described by various. Typical examples are the determination of a timeminimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. This paper demonstrates that methods commonly used to determine discrete approximations of probability distributions systematically underestimate the moments of the original distribution. Finite difference approximations for operators such as definite integration, interpolation, and differentiation are all special cases of linear functionals. Dual approximations in optimal control siam journal on. Discrete approximation an overview sciencedirect topics. Legendre pseudospectral approximations of optimal control problems i.
Optimal control problems for sweeping processes have been recently recognized among the most interesting and challenging problems in modern. Linearquadraticgaussian control, riccati equations, iterative linear approximations to nonlinear problems. Stationary points and global solutions of these approximating discrete time optimal control problems converge, as the discretization level is increased. The rst work dealt with the convergence of the optimal value or an optimal control for the discrete problem to the continuous solution see, e. A successive approximation approach designing optimal controller is developed for affine nonlinear discretetime systems with a quadratic performance index. Discrete approximation of linear functions from wolfram. Approximate maximum principle for discrete approximations of. Discrete approximations in optimal control springerlink. It has numerous applications in both science and engineering. Optimality models in motor control, promising research directions. An optimal control approach to deep learning and applications to discrete weight neural networks qianxiao li 1shuji hao abstract deep learning is formulated as a discrete time optimal control problem. Problem b determine the statecontrol function pair, 0, f.
Numerical analysis in optimal control springerlink. It is demonstrated that such a continuous problem can be replaced by a sequence of finite. It is shown that the resulting matrix riccati difference equation for closed. On the numerical treatment of linearquadratic optimal control problems for general linear timevarying differentialalgebraic equations. March 7, 2011 31 3 controllability, approximations, and optimal control 3. Equation for an optimal control problem of a di erentialalgebraic system j. Udc 6250 on difference approximations of optimal control systems pmmvol. Optimal control of a perturbed sweeping process via discrete. In this paper we explain and exemplify how one goes about analyzing the convergence of algorithms and discrete approximations in optimal control. An optimal control approach to deep learning and applications. Rungekutta integration is used to construct finitedimensional approximating problems that are consistent approximations, in the sense of polak 1993, to an original optimal control problem. Heemels abstract continuoustime linear constrained optimal control problems are in practice often solved using discretization techniques, e. An optimal control approach to deep learning and applications to discrete weight neural networks.
Pdf optimal control of a perturbed sweeping process via. In particular, we introduce the discretetime method of successive approximations msa, which is. Examples of stochastic dynamic programming problems. It is demonstrated that if p is a continuous optimal control problem whose system of differential equations is linear in the control and the state variables, and whose control and state variable constraint sets are convex, a direct method of determining an optimal solution of p exists. Rungekutta discretization of optimal control problems. Constrained state estimation for nonlinear discretetime. In this paper we consider a problem of minimization of a cost functional jm,a z t 0 z 1 0 e rtm fm. Frederic bonnans, philippe chartier, hasnaa zidani. Typical examples are the determination of a timeminimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal. Optimal control problems for sweeping processes have been recently recognized among the most interesting and challenging problems in modern control theory for discontinuous. Discrete approximations of probability distributions.
Mordukhovich 1 and ilya shvartsman 2 1 department of mathematics, wayne state university detroit, mi 48202, u. In this case equilibrium system solution of mfg is a critical point of an optimal control problem governed by transport equation. Stationary points and global solutions of these approximating discretetime optimal control problems converge, as the discretization level is increased. Direct collocation and nonlinear programming for optimal control problem using an enhanced transcribing scheme. The optimal control of a mechanical system is of crucial importance in many application areas. Convergence theory for a wide range of discretized optimal control problems is well established 16 17 48, except for some cases where, for example, the optimal control is of bang bang and. With the help of this approximation result, we show that the solution of the discrete lagrangian optimal control.
In this paper we present two techniques for analysis of discrete approximations in optimal control. Unlike the wellknown results for continuous plants, the closedform time optimal control for discrete time plants was never attained. It is demonstrated that such a continuous problem can be replaced by a sequence of finitedimensional. By using this approach the original optimal control problem is transformed into a sequence of nonhomogeneous linear twopoint boundary value tpbv problems. The ima volumes in mathematics and its applications, vol 78.
The examples thus far have shown continuous time systems and control solutions. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. An optimal control problem with discrete states and. A new procedure based on gaussian quadrature is developed in this paper. We consider the following formulation of an autonomous, mixed statecontrol constrained bolza optimal control problem with possibly free initial and terminal times. Convergence of discretetime approximations of constrained linearquadratic optimal control problems l. Discrete approximations of the hamiltonjacobi equation for. Costate approximation in optimal control using integral. Optimal control theory from a general perspective, an optimal control problem is an optimization problem. Deep learning is formulated as a discretetime optimal control problem. The link between mgf and optimal control takes place in the socalled potential case see 7 for the details. A singulaperturbation method is developed to obtain series solutions in terms of the outer, inner and intermediate series analogous to that in a continuous system.
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