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. Insection 3, we formulate the optimal dual control prob. Deep learning is formulated as a discrete time optimal control problem. In particular, a time optimal control law is constructed in. Lecture slides dynamic programming and stochastic control. Pdf the paper addresses an optimal control problem for a perturbed sweeping process of the rateindependent hysteresis type described by a controlled. Discrete approximations to continuous optimal control problems. Optimal control of a perturbed sweeping process via. Direct collocation and nonlinear programming for optimal control problem using an enhanced transcribing scheme.
The recent work of jingqing han sheds lights on this problem and is introduced. 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 from a general perspective, an optimal control problem is an optimization problem. Stationary points and global solutions of these approximating discretetime optimal control problems converge, as the discretization level is increased.
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. 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. An optimal control approach to deep learning and applications to discrete weight neural networks. Discrete approximations to continuous optimal control.
The constrained state estimation problem can be reformulated as a series of optimal control problems cf. Such a discretetime control system consists of four major parts. In this paper we explain and exemplify how one goes about analyzing the convergence of algorithms and discrete approximations in optimal control. Examples of stochastic dynamic programming problems. The main contribution of this paper is that the optimal control problems with. In particular, we introduce the discrete time method of successive approximations msa, which is. Discrete approximations to optimal trajectories using.
Solution of discretetime optimal control problems on. Discrete approximations of the hamiltonjacobi equation for an optimal control problem of a di erentialalgebraic system. Discrete approximation an overview sciencedirect topics. Linea rquadratic regulato r discrete optimal control discrete hamiltonjacobi eq. 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. An important class of continuoustime optimal control problems are the socalled linearquadratic optimal control problems where the objective functional j in 3. The difference between the two is that, in optimal control theory, the optimizer is a function, not just a single value. The link between mgf and optimal control takes place in the socalled potential case see 7 for the details. Discrete approximations to optimal control problems have been analyzed since the 1960s. In fact, as optimal control solutions are now often implemented digitally, contemporary control theory is now primarily concerned with discrete time systems and solutions.
In section 2 we study convergence properties of the optimal value and optimal solutions. Legendre pseudospectral approximations of optimal control. Discrete approximations of continuous distributions by maximum entropy economics letters, vol. Keywords sequential quadratic programming siam journal discrete approximation coercivity condition euler approximation. Discrete approximations to optimal trajectories using direct transcription and nonlinear programming. Constrained state estimation for nonlinear discretetime. Singularperturbation method for discrete models of. Discrete approximations of the hamiltonjacobi equation for. Optimal control problems for sweeping processes have been recently recognized among the most interesting and challenging problems in modern control theory for discontinuous.
On the numerical treatment of linearquadratic optimal control problems for general linear timevarying differentialalgebraic equations. Analysis of finite difference approximations of an optimal. Abstract pdf 3753 kb 1995 consistency of primaldual approximations for convex optimal control problems. It is demonstrated that such a continuous problem can be replaced by a sequence of finitedimensional. Error analysis of discrete approximations to bangbang. Convergence of discretetime approximations of constrained. It is demonstrated that such a continuous problem can be replaced by a sequence of finite. Discrete approximations of probability distributions. Optimality models in motor control, promising research directions. Unlike the wellknown results for continuous plants, the closedform time optimal control for discrete time plants was never attained. An optimal control problem with discrete states and. Solving the optimal control problems, however, is computationally demanding, because the problem dimension grows.
The closedloop and openloop optimal controls of a singularly perturbed continuous system are considered by means of their discrete models. Problem b determine the statecontrol function pair, 0, f. Proceedings of the 1999 ieee international symposium on computer aided control system design cat. This paper is devoted to optimal control of dynamical systems governed by differential inclusions in both frameworks of lipschitz continuous and discontinuous velocity mappings. The latter framework mostly concerns a new class of optimal control problems described by various.
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. Dual approximations in optimal control siam journal on. 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. Typical examples are the determination of a timeminimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal. Frederic bonnans, philippe chartier, hasnaa zidani to cite this version. We consider the following formulation of an autonomous, mixed statecontrol constrained bolza optimal control problem with possibly free initial and terminal times. 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. 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. An optimal control approach to deep learning and applications.
Deep learning is formulated as a discretetime optimal control problem. Numerical methods for solving optimal control problems. 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. It is shown that the resulting matrix riccati difference equation for closed. Discrete approximations in optimal control springerlink. Equation for an optimal control problem of a di erentialalgebraic system j.
Approximate maximum principle for discrete approximations of optimal control systems with nonsmooth objectives and endpoint constraints. The paper is devoted to the study of a new class of optimal control problems. Discrete approximations and optimal control of nonsmooth. Stationary points and global solutions of these approximating discrete time optimal control problems converge, as the discretization level is increased. Finiteapproximationerrorbased optimal control approach. Discrete approximations of the hamiltonjacobi equation. 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. 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. Convergence of discretetime approximations of constrained linearquadratic optimal control problems l. On difference approximations of optimal control systems.
Successive approximation approach of optimal control for. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to. Optimal recursive estimation, kalman lter, zakai equation. Discrete time optimal control problems, constrained optimization problem, parallel band solver, multiprocessor implementation, shared memory multiprocessor. Frederic bonnans, philippe chartier, hasnaa zidani. Discrete time optimal control applied to pest control problems. 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. Finite difference approximations for operators such as definite integration, interpolation, and differentiation are all special cases of linear functionals.
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. Heemels abstract continuoustime linear constrained optimal control problems are in practice often solved using discretization techniques, e. Discrete approximations to optimal trajectories using direct. Obtaining finite difference approximations using function values at equally spaced sample points is an important problem in numerical analysis. Optimization online optimal control of differential. Approximate maximum principle for discrete approximations. Linearquadraticgaussian control, riccati equations, iterative linear approximations to nonlinear problems. Optimal control problems for sweeping processes have been recently recognized among the most interesting and challenging. The examples thus far have shown continuous time systems and control solutions. The latter framework mostly concerns a new class of optimal control problems. Costate approximation in optimal control using integral. In this paper we present two techniques for analysis of discrete approximations in optimal control.
An optimal control approach to deep learning and applications to discreteweight neural networks. Approximate maximum principle for discrete approximations of. Rungekutta discretization of optimal control problems. Discretetime optimal control problems, constrained optimization problem, parallel band solver, multiprocessor implementation, shared memory multiprocessor. Discrete approximations of the hamiltonjacobi equation for an optimal control problem of a di erentialalgebraic system j. The term w t w to will be normally distributed with mean zero and variance t t 0. In particular, we introduce the discretetime method of successive approximations msa, which is. Proceedings of the 35th international conference on machine learning, in pmlr 80. According to various authors, an optimal control problem which is representable by a system of ordinary differential equations, an integral cost function, and control and state variable constraints is a continuous optimal control problem. This allows one to characterize necessary conditions for optimality and develop training algorithms that do not rely on gra. March 7, 2011 31 3 controllability, approximations, and optimal control 3. Mordukhovich 1 and ilya shvartsman 2 1 department of mathematics, wayne state university detroit, mi 48202, u. Discrete approximation of linear functions from wolfram.
Legendre pseudospectral approximations of optimal control problems 3. Udc 6250 on difference approximations of optimal control systems pmmvol. Discrete hamiltonjacobi theory and discrete optimal control. The optimal control of a mechanical system is of crucial importance in many application areas. Duality of optimal control and optimal estimation including new results. Optimal control problems for sweeping processes have been recently recognized among the most interesting and challenging problems in modern. In this case equilibrium system solution of mfg is a critical point of an optimal control problem governed by transport equation. T and y 2 rn is said to be the optimal value function. 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. It has numerous applications in both science and engineering. View enhanced pdf access article on wiley online library html view download pdf for offline viewing. Pdf optimal control of a perturbed sweeping process via. Discrete time optimal control applied to pest control problems 481 the paper is organized as follows. Legendre pseudospectral approximations of optimal control problems i.
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. In this paper we consider a problem of minimization of a cost functional jm,a z t 0 z 1 0 e rtm fm. The algorithm presented here solved the approximation problem for an arbitrary linear functional. A successive approximation approach designing optimal controller is developed for affine nonlinear discretetime systems with a quadratic performance index. The ima volumes in mathematics and its applications, vol 78. Discretetime linear systems discretetime linear systems discretetime linear system 8 pdf 75 kb. 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. A new procedure based on gaussian quadrature is developed in this paper.
This paper demonstrates that methods commonly used to determine discrete approximations of probability distributions systematically underestimate the moments of the original distribution. By using this approach the original optimal control problem is transformed into a sequence of nonhomogeneous linear twopoint boundary value tpbv problems. Optimal control theory is a mature mathematical discipline with numerous applications. This problem can be reduced to dynamic optimization of a stateconstrained unbounded differential inclusion with highly irregular data that cannot be treated. Optimal control of a perturbed sweeping process via discrete. Numerical analysis in optimal control springerlink. Optimization online optimal control of differential inclusions. With the help of this approximation result, we show that the solution of the discrete lagrangian optimal control. 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.