Stable Numerical Schemes for Solving Hamilton–Jacobi–Bellman–Isaacs Equations

Botkin, Nikolai D.; Hoffmann, Karl-Heinz; Turova, Varvara

doi:10.1137/100801068

Cited by 39 publications

(39 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let us assume that the target is a compact subset of R d with non empty interior and piecewise smooth boundary. The major difficulty dealing with this problem is that the time of arrival to the target starting from the point x t(x, α(·)) := inf 8) can be infinite at some points. As a consequence, the minimum time function defined as…”

mentioning

confidence: 99%

An Efficient Policy Iteration Algorithm for Dynamic Programming Equations

Alla¹,

Falcone²,

Kalise³

2015

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Abstract. We present an accelerated algorithm for the solution of static Hamilton-JacobiBellman equations related to optimal control problems. Our scheme is based on a classic policy iteration procedure, which is known to have superlinear convergence in many relevant cases provided the initial guess is sufficiently close to the solution. This limitation often degenerates into a behavior similar to a value iteration method, with an increased computation time. The new scheme circumvents this problem by combining the advantages of both algorithms with an efficient coupling. The method starts with a coarse-mesh value iteration phase and then switches to a fine-mesh policy iteration procedure when a certain error threshold is reached. A delicate point is to determine this threshold in order to avoid cumbersome computations with the value iteration and, at the same time, to be reasonably sure that the policy iteration method will finally converge to the optimal solution. We analyze the methods and efficient coupling in a number of examples in dimension two, three and four illustrating their properties.

show abstract

mentioning

confidence: 99%

An Efficient Policy Iteration Algorithm for Dynamic Programming Equations

Alla¹,

Falcone²,

Kalise³

2015

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

“…Notice that Condition (i) provides the embedding of the level sets of V into the corresponding level sets of g. Condition (ii) provides the u-stability of functions V (see [15,16]), and therefore, the u-stability of the function V . The operation "inf" provides the minimality of the resulting function, i.e., the maximality of its level sets.…”

Section: Numerical Schemementioning

confidence: 99%

“…On the other hand, Theorem 1 shows that the function V can be computed as lim t→−∞ V (t, ·), where V (t, x) is the value function of the differential game with the Hamiltonian H(x, p) and the objective functional J(x(·)) = max τ ∈[t,0] {x(0), g(τ )}; see [16]. This remark allows us to use the numerical methods developed for constructing time-dependent value functions in differential games with state constraints (see [15,16]). …”

Section: Numerical Schemementioning

confidence: 99%

See 1 more Smart Citation

Numerical Construction of Viable Sets for Autonomous Conflict Control Systems

Botkin¹,

Turova²

2014

Mathematics

View full text Add to dashboard Cite

A conflict control system with state constraints is under consideration. A method for finding viability kernels (the largest subsets of state constraints where the system can be confined) is proposed. The method is related to differential games theory essentially developed by N. N. Krasovskii and A. I. Subbotin. The viability kernel is constructed as the limit of sets generated by a Pontryagin-like backward procedure. This method is implemented in the framework of a level set technique based on the computation of limiting viscosity solutions of an appropriate Hamilton-Jacobi equation. To fulfill this, the authors adapt their numerical methods formerly developed for solving time-dependent Hamilton-Jacobi equations arising from problems with state constraints. Examples of computing viability sets are given.

show abstract

“…[2,4,14,24]. In papers [7,8], the Hamilton-Jacobi equation approach is extended to more general cost functionals.…”

Section: Introductionmentioning

confidence: 99%

Aircraft Control During Cruise Flight in Windshear Conditions: Viability Approach

et al. 2017

View full text Add to dashboard Cite

This paper addresses the analysis of aircraft control capabilities during the cruise phase (flying at the established level with practically constant configuration and speed) in the presence of windshears. The study uses a point-mass aircraft model describing flight in a vertical plane. The problem is formulated as a differential game against wind disturbances. The first player, autopilot, controls the angle of attack and the power setting, whereas the second player, wind, produces dangerous gusts. The state variables of the model are subjected to constraints expressing aircraft safety conditions. Namely, the altitude, path inclination, and velocity are constrained. Viability theory is used to find the so-called viability kernel, the maximal subset of the state constraint where the aircraft trajectories can remain arbitrary long if the first player utilizes an appropriate feedback control, and the second player generates any admissible disturbances. The computations are based on grid methods developed by the authors and implemented on a multiprocessor computer system.

show abstract

Stable Numerical Schemes for Solving Hamilton–Jacobi–Bellman–Isaacs Equations

Cited by 39 publications

References 14 publications

An Efficient Policy Iteration Algorithm for Dynamic Programming Equations

An Efficient Policy Iteration Algorithm for Dynamic Programming Equations

Numerical Construction of Viable Sets for Autonomous Conflict Control Systems

Aircraft Control During Cruise Flight in Windshear Conditions: Viability Approach

Contact Info

Product

Resources

About