Approximation schemes for solving disturbed control problems with non-terminal time and state constraints

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

doi:10.1524/anly.2011.1122

Cited by 25 publications

(34 citation statements)

References 11 publications

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“…A grid approximation of such a function can be computed as a limiting solution, as t → −∞, of an appropriate Hamilton-Jacobi equation arising from conflict control problems with state constraints (see [7]). The algorithm looks as follows (cf.…”

Section: Grid Methods For Computing Viability Kernelsmentioning

confidence: 99%

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Aircraft Control During Cruise Flight in Windshear Conditions: Viability Approach

et al. 2017

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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.

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Section: Grid Methods For Computing Viability Kernelsmentioning

confidence: 99%

“…The proof follows from the fact that the both operators (33) and (34) satisfy the same consistency condition (see [7]) corresponding to the Hamiltonian…”

Section: Remarkmentioning

confidence: 98%

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Aircraft Control During Cruise Flight in Windshear Conditions: Viability Approach

et al. 2017

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“…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%

Numerical Construction of Viable Sets for Autonomous Conflict Control Systems

Botkin¹,

Turova²

2014

Mathematics

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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.

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Quick Construction of Dangerous Disturbances in Conflict Control Problems

Martynov

Botkin

Turova

et al. 2020

Annals of the International Society of Dynamic Games

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The paper is devoted to the construction of dangerous disturbances in linear conflict control problems. Using the technique of sequential linearization, dangerous disturbances can also be constructed for nonlinear systems such as aircraft dynamics equations, including filters, servomechanisms, etc. The procedure proposed is based on a dynamic programming method and consists in the backward integration of ordinary matrix differential equations defining centers, sizes, and orientations of time-dependent parallelotopes forming a repulsive tube in the time-space domain. A feedback disturbance strategy can keep the state vector of the conflict control system outside the repulsive tube for all admissible inputs of the control.

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Approximation schemes for solving disturbed control problems with non-terminal time and state constraints

Cited by 25 publications

References 11 publications

Aircraft Control During Cruise Flight in Windshear Conditions: Viability Approach

Aircraft Control During Cruise Flight in Windshear Conditions: Viability Approach

Numerical Construction of Viable Sets for Autonomous Conflict Control Systems

Quick Construction of Dangerous Disturbances in Conflict Control Problems

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