On the structure of locally Lipschitz minimax solutions of the Hamilton-Jacobi-Bellman equation in terms of classical characteristics

Subbotina, N. N.; Kolpakova, E. A.

doi:10.1134/s0081543810050160

Cited by 6 publications

(3 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The boundary value problems for characteristic systems of ordinary differential equations (describing the dynamics of the state and adjoint variables along characteristic curves) were involved there. However, the related computational algorithms as formulated in [45][46][47] are rather complicated and also suffer from the curse of dimensionality.…”

Section: Introductionmentioning

confidence: 99%

Perspectives on Characteristics Based Curse-of-Dimensionality-Free Numerical Approaches for Solving Hamilton–Jacobi Equations

Yegorov

Dower

2018

Appl Math Optim

View full text Add to dashboard Cite

This paper extends the considerations of the works [1,2] regarding curse-of-dimensionality-free numerical approaches to solve certain types of Hamilton-Jacobi equations arising in optimal control problems, differential games and elsewhere. A rigorous formulation and justification for the extended Hopf-Lax formula of [2] is provided together with novel theoretical and practical discussions including useful recommendations. By using the method of characteristics, the solutions of some problem classes under convexity/concavity conditions on Hamiltonians (in particular, the solutions of Hamilton-Jacobi-Bellman equations in optimal control problems) are evaluated separately at different initial positions. This allows for the avoidance of the curse of dimensionality, as well as for choosing arbitrary computational regions. The corresponding feedback control strategies are obtained at selected positions without approximating the partial derivatives of the solutions. The results of numerical simulations demonstrate the high potential of the proposed techniques. It is also pointed out that, despite the indicated advantages, the related approaches still have a limited range of applicability, and their extensions to Hamilton-Jacobi-Isaacs equations in zero-sum two-player differential games are currently developed only for sufficiently narrow classes of control systems. That is why further extensions are worth investigating.

show abstract

Section: Introductionmentioning

confidence: 99%

Perspectives on Characteristics Based Curse-of-Dimensionality-Free Numerical Approaches for Solving Hamilton–Jacobi Equations

Yegorov

Dower

2018

Appl Math Optim

View full text Add to dashboard Cite

show abstract

“…Recall that admissible integral trajectories of the controlled system and in inverse time τ : = T − t fulfilling Pontryagin's maximum principle are called characteristics of the problem . Also note that the superdifferential

D_{(c MathClass-punc, y MathClass-punc, h MathClass-punc, τ)}^{MathClass-bin+} S (c MathClass-rel' MathClass-punc, y MathClass-rel' MathClass-punc, h MathClass-rel' MathClass-punc, τ MathClass-rel')

of the Bellman function S at a point( c ′ , y ′ , h ′ , τ ′ ) ∈ G × [0, T ] coincides with the convex hull of all the vectors

((), MathClass-bin- ψ_{1} (t MathClass-rel') MathClass-punc, MathClass-bin- ψ_{2} (t MathClass-rel') MathClass-punc, MathClass-bin- ψ_{3} (t MathClass-rel') MathClass-punc, scriptH (c (t MathClass-rel') MathClass-punc, y (t MathClass-rel') MathClass-punc, h (t MathClass-rel') MathClass-punc, ψ_{1} (t MathClass-rel') MathClass-punc, ψ_{2} (t MathClass-rel') MathClass-punc, ψ_{3} (t MathClass-rel'))) MathClass-punc, t MathClass-rel' MathClass-punc: MathClass-rel= T MathClass-bin- τ MathClass-rel' MathClass-punc,

…”

Section: Synthesis Of Optimal Controlmentioning

confidence: 99%

Synthesis of optimal control in a mathematical model of tumour–immune dynamics

Yegorov

Todorov

2013

Optim Control Appl Methods

View full text Add to dashboard Cite

SummaryAn optimal control problem for a mathematical model of tumour–immune dynamics under the influence of chemotherapy is considered. The toxicity effect of the chemotherapeutic agent on both tumour and immunocompetent cells is taken into account. A standard linear pharmacokinetic equation for the chemotherapeutic agent is added to the system. The aim is to find an optimal strategy of treatment to minimize the tumour volume while keeping the immune response not lower than a fixed permissible level as far as possible. Sufficient conditions for the existence of not more than one switching and not more than two switchings without singular regimes are obtained. The surfaces in the extended phase space, on which the last switching appears, are constructed analytically.Copyright © 2013 John Wiley & Sons, Ltd.

show abstract

“…The following assertion on the connection of Definition 2 with the definitions of the minimax solution and the viscosity solution is a consequence of results in [4,5,[10][11][12]. …”

Section: Generalized Solution To Problem (11)mentioning

confidence: 99%

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

Rodin

2016

UMJ

View full text Add to dashboard Cite

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.

show abstract

On the structure of locally Lipschitz minimax solutions of the Hamilton-Jacobi-Bellman equation in terms of classical characteristics

Cited by 6 publications

References 4 publications

Perspectives on Characteristics Based Curse-of-Dimensionality-Free Numerical Approaches for Solving Hamilton–Jacobi Equations

Perspectives on Characteristics Based Curse-of-Dimensionality-Free Numerical Approaches for Solving Hamilton–Jacobi Equations

Synthesis of optimal control in a mathematical model of tumour–immune dynamics

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

Contact Info

Product

Resources

About