UNIQUENESS OF VISCOSITY SOLUTIONS OF STOCHASTIC HAMILTON-JACOBI EQUATIONS

This article is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi(HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the associated stochastic HJ equation.     

Bounded-from-below solutions of the Hamilton-Jacobi equation for optimal control problems with exit times: vanishing lagrangians,eikonal equations,and shape-from-shading

We study the Hamilton-Jacobi equation for undiscounted exit time control problems with general nonnegative Lagrangians using the dynamic programming approach. We prove theorems characterizing the value function as the unique bounded-from-below viscosity solution of the Hamilton-Jacobi equation that is null on the target. The result applies to problems with the property that all trajectories satisfying a certain integral condition must stay in a bounded set. We allow problems for which the Lagrangian is not uniformly bounded below by positive constants, in which the hypotheses of the known uniqueness results for Hamilton-Jacobi equations are not satisfied. We apply our theorems to eikonal equations from geometric optics, shape-from-shading equations from image processing, and variants of the Fuller Problem.     

Hamilton-Jacobi theory over time scales and applications to linear-quadratic problems

In this paper we first derive the verification theorem for nonlinear optimal control problems over time scales. That is, we show that the value function is the only solution of the Hamilton-Jacobi equation, in which the minimum is attained at an optimal feedback controller. Applications to the linear-quadratic regulator problem (LQR problem) gives a feedback optimal controller form in terms of the solution of a generalized time scale Riccati equation, and that every optimal solution of the LQR problem must take that form. A connection of the newly obtained Riccati equation with the traditional one is established. Problems with shift in the state variable are also considered. As an important tool for the latter theory we obtain a new formula for the chain rule on time scales. Finally, the corresponding LQR problem with shift in the state variable is analyzed and the results are related to previous ones.     

On viscosity solution of functional Hamilton-Jacobi type equations for hereditary systems

The paper is devoted to the development of the viscosity approach to the generalized solution of functional Hamilton-Jacobi type equations with coinvariant derivatives and a nonanticipatory Hamiltonian. These equations are naturally connected to problems of dynamical optimization of hereditary systems and, as compared with classical Hamilton-Jacobi equations, possess a number of additional peculiarities stipulated by the aftereffect. The definition of a viscosity solution that takes the above peculiarities into account is given. The consistency of this definition with the notion of a classical solution and with the minimax approach to the generalized solution is substantiated. The existence and uniqueness theorems are proved.     

有限时区非线性系统的最优切换控制

This paper proposes a optimal control problem for a general nonlinear systems with finitely many admissible control settings and with costs assigned to switching of controls. With dynamic programming and viscosity solution theory we show that the switching lower-value function is a viscosity solution of the appropriate systems of quasi-variational inequalities(the appropriate generalization of the Hamilton-Jacobi equation in this context) and that the minimal such switching-storage function is equal to the continuous switching lower-value for the game. With the lower value function a optimal switching control is designed for minimizing the cost of running the systems.     

The vanishing viscosity method for the sensitivity analysis of an optimal control problem of conservation laws in the presence of shocks

In this article we study, by the vanishing viscosity method, the sensitivity analysis of an optimal control problem for 1-D scalar conservation laws in the presence of shocks. It is reduced to investigate the vanishing viscosity limit for the nonlinear conservation law, the corresponding linearized equation and its adjoint equation, respectively. We employ the method of matched asymptotic expansions to construct approximate solutions to those equations. It is then proved that the approximate solutions, respectively, satisfy those viscous equations in the asymptotic sense, and converge to the solutions of the corresponding inviscid problems with certain convergent rates. A new equation for the variation of shock positions is derived. It is also discussed how to identify descent directions to find the minimizer of the viscous optimal control problem in the quasi-shock case.     

半线性热方程的反馈零能控性

考虑具有Lipschitz非线性项,半线性热方程的最优控制问题.我们将运用观测不等式,证明值函数ψ作为相应Hamilton-Jacobi方程的唯一粘性正解是局部Lipschitz连续的.最后,运用动态规划方法,得到系统最优的反馈控制.     

Minimax and viscosity solutions in optimization problems for hereditary systems

For a dynamical system with discrete and distributed time delays, a control problem under disturbance or counteraction is considered. The problem is formalized in the context of the game-theoretical approach in the class of control strategies with memory. The problem is associated with a functional Hamilton-Jacobi type equation with coinvariant derivatives. The minimax and viscosity approaches to a generalized solution to this equation are discussed. It is shown that, under the same condition at the right endpoint, the minimax and viscosity solutions coincide, thereby uniquely defining the functional of optimal guaranteed result in the control problem.     

Exponential Convergence to Time-Periodic Viscosity Solutions in Time-Periodic Hamilton-Jacobi Equations

Consider the Cauchy problem of a time-periodic Hamilton-Jacobi equation on a closed manifold, where the Hamiltonian satisfies the condition: The Aubry set of the corresponding Hamiltonian system consists of one hyperbolic 1-periodic orbit. It is proved that the unique viscosity solution of Cauchy problem converges exponentially fast to a 1-periodic viscosity solution of the Hamilton-Jacobi equation as the time tends to infinity.     

Hamilton-Jacobi Equations with Discontinuous Source Terms

We study the initial-value problem for a Hamilton-Jacobi equation whose Hamiltonian is discontinuous with respect to state variables. Our motivation comes from a model describing the two dimensional nucleation in crystal growth phenomena. A typical equation has a semicontinuous source term. We introduce a new notion of viscosity solutions and prove among other results that the initial-value problem admits a unique global-in-time uniformly continuous solution for any bounded uniformly continuous initial data. We also give a representation formula of the solution as a value function by the optimal control theory with a semicontinuous running cost function.     

Optimal control problems on manifolds: a dynamic programming approach

Dynamic programming identifies the value function of continuous time optimal control with a solution to the Hamilton-Jacobi equation, appropriately defined. This relationship in turn leads to sufficient conditions of global optimality, which have been widely used to confirm the optimality of putative minimisers. In continuous time optimal control, the dynamic programming methodology has been used for problems with state space a vector space. However there are many problems of interest in which it is necessary to regard the state space as a manifold. This paper extends dynamic programming to cover problems in which the state space is a general finite-dimensional C^∞ manifold. It shows that, also in a manifold setting, we can characterise the value function of a free time optimal control problem as a unique lower semicontinuous, lower bounded, generalised solution of the Hamilton-Jacobi equation. The application of these results is illustrated by the investigation of minimum time controllers for a rigid pendulum.     

On the Bellman Equation for Infinite Horizon Problems with Unbounded Cost Functional

We study a class of infinite horizon control problems for nonlinear systems, which includes the Linear Quadratic (LQ) problem, using the Dynamic Programming approach. Sufficient conditions for the regularity of the value function are given. The value function is compared with sub- and supersolutions of the Bellman equation and a uniqueness theorem is proved for this equation among locally Lipschitz functions bounded below. As an application it is shown that an optimal control for the LQ problem is nearly optimal for a large class of small unbounded nonlinear and nonquadratic pertubations of the same problem. Accepted 8 October 1998     

Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost

We study viscosity solutions of Hamilton-Jacobi equations that arise in optimal control problems with unbounded controls and discontinuous Lagrangian. In our assumptions, the comparison principle will not hold, in general. We prove optimality principles that extend the scope of the results of [23] under very general assumptions, allowing unbounded controls. In particular, our results apply to calculus of variations problems under Tonelli type coercivity conditions. Optimality principles can be applied to obtain necessary and sufficient conditions for uniqueness in boundary value problems, and to characterize minimal and maximal solutions when uniqueness fails. We give examples of applications of our results in this direction.     

Application of viscosity solutions of infinite-dimensional Hamilton-Jacobi-Bellman equations to some problems in distributed optimal control

We apply the recently developed Crandall and Lions theory of viscosity solutions for infinite-dimensional Hamilton-Jacobi equations to two problems in distributed control. The first problem is governed by differential-difference equations as dynamics, and the second problem is governed by a nonlinear divergence form parabolic equation. We prove a Pontryagin maximum principle in each case by deriving the Bellman equation and using the fact that the value function is a viscosity supersolution.This work was supported by the Air Force Office for Scientific Research, Grant No. AFOSR-86-0202. The author would like to thank R. Jensen for several helpful conversations regarding the problems discussed here. He would also like to thank M. Crandall for providing early preprints of his work in progress with P. L. Lions on infinite-dimensional problems.     

Uniqueness for first-order Hamilton-Jacobi equations and Hopf formula

We study uniqueness properties for a certain class of Cauchy problems for first-order Hamilton-Jacobi equations for which a solution is given by the Hopf formula. We prove various comparison and characterisation results concerning both convex generalized solutions and viscosity solutions. In particular, we show that the Hopf solution is the maximum convex generalized subsolution and the unique convex viscosity solution of the Cauchy problem.     

Infinite horizon optimal control problems with multiple thermostatic hybrid dynamics

We study an optimal control problem for a hybrid system exhibiting several internal switching variables whose discrete evolutions are governed by some delayed thermostatic laws. By the dynamic programming technique we prove that the value function is the unique viscosity solution of a system of several Hamilton-Jacobi equations, suitably coupled. The method involves a contraction principle and some suitably adapted results for exit-time problems with discontinuous exit cost.     

两端固定的奇异梁方程的多重正解

设 n 是一个任意的自然数. 证明了一个两端固定的奇异梁方程的 n 个正解的存在性, 其中非线性项是一个Carathéodory 函数. 主要工具是涉及非线性项的高度函数与锥压缩锥拉伸型的 Krasnoselskii不动点定理. 进一步的研究表明,如果非线性项在零点和无穷远处的增长极限均为无界函数, 该方程仍可能具有正解.     

Extremal points and optimal control theory

Summary An optimal control problem is considered in a setting akin to that of the theory. of generalized curves. Rather than minimizing a functional depending on pairs of trajectories and controls subject to some constraints, a functional defined on a set of Radon measures is considered; the set of measures is determined by the constraints. An approximation scheme is developed, so that the solution of the optimal control problems can be effected by solving a sequence of nonlinear programming problems. Several existence theorems for this kind of generalized control problems are then proved; the most interesting is the one concerning problems in which the set of allowable controls is unbounded. Entrata in Redazione il 5 febbraio 1975.     

THE RELAXING SCHEMES FOR HAMILION-JACOBI EQUATIONS

1. IntroductionWe are illterested in the numerical approximation of viscosity solution of the following lirstorder Hamilton-Jacobi eqllationopt + H(&.,, rk..' ...) gb'.) - 0' (1'1)with initial data ac(~, 0) = ado(x). It is well known that the solutions to problem (1.1) typicallyare continuous (typically they are locally Lipschitz continuous) but with discoatinuous derivatives, even though the initial data ado E Coo. The nonuniqueness of such solutions to (1.1) alsonecessitates the introducti…     

An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

An initial boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann–Dirichlet type is investigated in this work. From a physical point of view, the initial boundary value problem considered here is a mathematical model of quasistationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. On the basis of a priori estimates, we prove a local existence theorem and uniqueness for a weak generalized solution of the initial boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow-up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow-up of a solution in a finite time in a limited area with a nonlinear Neumann–Dirichlet boundary condition are obtained.