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Discontinuous solutions of the Hamilton-Jacobi equation for exit time problems

SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2000年第4期38卷 1067-1085页

作者： Ye, JJ Univ Victoria Dept Math & Stat Victoria BC V8W 3P4 Canada

In general, the value function associated with an exit time problem is a discontinuous function. We prove that the lower (upper) semicontinuous envelope of the value function is a supersolution (subsolution) of the Hamilton Jacobi equation involving the proximal subdifferentials (superdifferentials) with subdifferential-type (superdifferential-type) mixed boundary condition. We also show that if the value function is upper semicontinuous, then it is the maximum subsolution of the Hamilton Jacobi equation involving the proximal superdifferentials with the natural boundary condition, and if the value function is lower semicontinuous, then it is the minimum solution of the Hamilton Jacobi equation involving the proximal subdifferentials with a natural boundary condition. Futhermore, if a compatibility condition is satis ed, then the value function is the unique lower semicontinuous solution of the Hamilton Jacobi equation with a natural boundary condition and a subdifferential type boundary condition. Some conditions ensuring lower semicontinuity of the value functions are also given.

关键词： Hamilton-Jacobi equation dynamic programming principle exit time problems proximal subdifferentials

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Deterministic exit time control problems with discontinuous exit costs

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1997年第2期35卷 399-434页

作者： Blanc, AP Fac. des Sciences et Techniques Université de Tours Parc de Grandmont 37200 Tours France

We study deterministic exit time control problems with discontinuous exit costs. When the exit cost phi is upper semicontinuous and there is an outer field on the boundary, we show that all the value functions have the same lower semicontinuous envelope which is the unique lower semicontinuous viscosity solution of the associated Dirichlet problem. We also prove uniqueness results for the generalized Dirichlet problem for first-order Hamilton-Jacobi equations with convex Hamiltonians and with discontinuous boundary conditions, under some nondegeneracy conditions on the Hamiltonians on the boundary.

关键词： exit time problems Hamilton-Jacobi equations Dirichlet problems discontinuous data viscosity solutions

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

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NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS 2004年第1期11卷 95-122页

作者： Malisoff, M Louisiana State Univ Dept Math Baton Rouge LA 70803 USA

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 Puller Problem.

关键词： optimal control dynamic programming viscosity solutions exit time problems

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Singular perturbation of a finite horizon problem with state-space constraints

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1998年第6期36卷 2040-2060页

作者： Bagagiolo, F Bardi, M Univ Trent Dipartimento Matemat I-38050 Povo Trento Italy Univ Padua Dipartimento Matemat I-35131 Padua Italy

We study the singular perturbation of optimal control problems for nonlinear systems with constraints on the fast state variables and a cost functional either of Bolza type or involving the exit time of the system from a given domain. Under a controllability assumption on the fast variables, we show that these variables become controls in the limit problem. Our method consists of passing to the limit in the associated Hamilton-Jacobi-Bellman (HJB) equations by means of some tools in the theory of viscosity solutions.

关键词： singular perturbations optimal control nonlinear systems viscosity solutions Hamilton-Jacobi equations Bolza problem exit time problems state constraints

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Comparison principle for the Cauchy problem for Hamilton-Jacobi equations with discontinuous data

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2001年第8期45卷 1015-1037页

作者： Blanc, AP IACM FORTH Iraklion 71110 Crete Greece

The results between discontinuous viscosity solutions of initial-boundary value problems with discontinuous initial and Dirichlet conditions for first-order Hamilton-Jacobi equations were compared. The applications of these results to optimal exit time problems and to the convergence of their numerical approximations were considered. The numerical scheme of an optimal control problem was given.

关键词： evolutive Hamilton-Jacobi equations initial and boundary value problems discontinuous data exit time problems viscosity solutions

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Drift Counteraction and Control of Downsized and Underactuated Systems: What MPC Has to Offer?

Drift Counteraction and Control of Downsized and Underactuat...

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6th International-Federation-of-Automatic-Control (IFAC) Conference on Nonlinear-Model-Predictive-Control (NMPC)

作者： Kolmanovsky, Ilya Zidek, Robert Univ Michigan Dept Aerosp Engn Ann Arbor MI 48109 USA

Motivated by practical applications, the paper highlights MPC formulations suitable for counteracting system drift and the effects of large measured disturbances/set-point changes, and for overcoming limitations of underactuation. For drift counteraction, where the objective is to maximize the time or yield until the system trajectory exits a prescribed set, defined by system safety constraints, operating limits and/or efficiency requirements, MPC, based on mixed integer or conventional linear programming, can lead to effective solutions for higher order systems than possible with dynamic programming and value iterations based methods. Such solutions can have broad applicability including for fuel optimal Geostationary Orbit (GEO) station keeping, spacecraft Low Earth Orbit (LEO) maintenance, underactuated spacecraft attitude control, hybrid electric propulsion energy management, glider flight management, and the development of driving policies for adaptive cruise control and autonomous driving. The paper also highlights the capability of MPC to achieve non-smooth local stabilization of underactuated systems that can be exploited for attitude control of underactuated spacecraft with reaction wheel failures. Related results for systems that cannot be globally stabilized by continuous feedback are also mentioned. Finally, opportunities to handle large load and set-point changes with reference governors are discussed. (C) 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

关键词： Constrained Control Model Predictive Control exit time problems Optimal Stopping Underactuated Systems Reference Governors

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