First Order PDEs in Variation Calculus, Optimal Control and Differential Games

Melikyan, Arik

doi:10.1007/978-1-4612-1758-9_4

Cited by 30 publications

(62 citation statements)

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“…One of the close works in this area is the book by A.A. Melikyan [6], where partial differential equations of the first order. Where considered in the class of continuously differentiable and piecewise smooth input data.…”

Section: Introductionmentioning

confidence: 99%

“…His work is mainly devoted to the issue of development of the method of characteristics and its further application to constructions of solutions in the following cases: a) the generalized viscosity solution is not smooth and then the Hamiltonian is smooth or non-smooth function; b) the solution is smooth, but the Hamiltonian is non smooth. The Poisson brackets are the main tools in the study [6].…”

Section: Introductionmentioning

confidence: 99%

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On the Structure of the Singular Set of a Piecewise Smooth Minimax Solution of the Hamilton–jacobi–bellman Equation

Rodin

2016

UMJ

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The properties of a minimax piecewise smooth solution of the Hamilton-Jacobi-Bellman equation are studied. We get a generalization of the nesessary and sufficient conditions for the points of nondifferentiability (singularity) of the minimax solution and the Rankine-Hugoniot condition. We describe the dimensions of smooth manifolds containing in the singular set of the piecewise smooth solution in terms of state characteristics crossing on this singular set. New structural properties of the singular set are obtained for the case of the Hamiltonian depending only on the impulse variable.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the Structure of the Singular Set of a Piecewise Smooth Minimax Solution of the Hamilton–jacobi–bellman Equation

Rodin

2016

UMJ

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“…The second inequality (4.26) follows from Melikyan's [M,Theorem 1.6] on the solvability of irregular, noncharacteristic problems for nonlinear first-order PDE.…”

Section: Introduce Finally the Expressions A(t) := K(q(t)) B(t) := (Qmentioning

confidence: 99%

Envelopes and nonconvex Hamilton–Jacobi equations

Evans

2013

Calc. Var.

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“…In [15], a pursuit-evasion game has been analyzed with the pursuer and the evader constrained to move on a twodimensional conical surface in a three-dimensional space. A theoretical framework based on the method of characteristics has been presented in [14] to address such problems. The inherent hardness in obtaining an analytical solution to the associated Hamilton-Jacobi-Isaacs equation has led to the development of numerical techniques for the computation of the value function.…”

Section: Introductionmentioning

confidence: 99%

Singular surfaces in multi-agent connectivity maintenance games

Bhattacharya

Başar

Hovakimyan

2011

IEEE Conference on Decision and Control and European Control Conference

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Abstract-In this paper, we analyze a connectivity maintenance problem that arises when two mobile autonomous agents navigate in an environment containing obstacles. Using the Rician fading model for the communication channel, we address the problem for the case when the motion strategies for one of the agents is adversarial in nature. We investigate a specific kind of singular surface that appears in the solution to the underlying pursuit-evasion game, namely the dispersal surface. We present construction of the projection of several dispersal surfaces for various obstacle geometries by fixing the initial position of the evader. Finally, we present numerical simulations for specific environments containing obstacles.

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First Order PDEs in Variation Calculus, Optimal Control and Differential Games

Cited by 30 publications

References 0 publications

On the Structure of the Singular Set of a Piecewise Smooth Minimax Solution of the Hamilton–jacobi–bellman Equation

On the Structure of the Singular Set of a Piecewise Smooth Minimax Solution of the Hamilton–jacobi–bellman Equation

Envelopes and nonconvex Hamilton–Jacobi equations

Singular surfaces in multi-agent connectivity maintenance games

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