scholarly journals Compatibility of state constraints and dynamics for multiagent control systems

2021 ◽  
Author(s):  
Giulia Cavagnari ◽  
Antonio Marigonda ◽  
Marc Quincampoix
Keyword(s):  
Continuity Equation ◽  
State Constraints ◽  
State Variable ◽  
Constraint Set ◽  
Viscosity Sense ◽  
Multiagent Control ◽  
Wasserstein Space ◽  
Hamilton Jacobi Bellman ◽  
Necessary And Sufficient ◽  
Jacobi Equations

AbstractThis study concerns the problem of compatibility of state constraints with a multiagent control system. Such a system deals with a number of agents so large that only a statistical description is available. For this reason, the state variable is described by a probability measure on $${\mathbb {R}}^d$$ R d representing the density of the agents and evolving according to the so-called continuity equation which is an equation stated in the Wasserstein space of probability measures. The aim of the paper is to provide a necessary and sufficient condition for a given constraint (a closed subset of the Wasserstein space) to be compatible with the controlled continuity equation. This new condition is characterized in a viscosity sense as follows: the distance function to the constraint set is a viscosity supersolution of a suitable Hamilton–Jacobi–Bellman equation stated on the Wasserstein space. As a byproduct and key ingredient of our approach, we obtain a new comparison theorem for evolutionary Hamilton–Jacobi equations in the Wasserstein space.

Two-player zero-sum stochastic differential games with random horizon

2019 ◽  
Vol 51 (4) ◽  
pp. 1209-1235
Author(s):  
M. Ferreira ◽  
D. Pinheiro ◽  
S. Pinheiro
Keyword(s):  
Viscosity Solution ◽  
Nonlinear Partial Differential Equation ◽  
Random Variable ◽  
Planning Horizon ◽  
Continuous Random Variable ◽  
State Variable ◽  
Random Horizon ◽  
Hamilton Jacobi Bellman ◽  
Zero Sum ◽  
Random Planning Horizon

AbstractWe consider a two-player zero-sum stochastic differential game with a random planning horizon and diffusive state variable dynamics. The random planning horizon is a function of a non-negative continuous random variable, which is assumed to be independent of the Brownian motion driving the state variable dynamics. We study this game using a combination of dynamic programming and viscosity solution techniques. Under some mild assumptions, we prove that the value of the game exists and is the unique viscosity solution of a certain nonlinear partial differential equation of Hamilton–Jacobi–Bellman–Isaacs type.

Solving the Hamilton-Jacobi-Bellman equation of stochastic control by a semigroup perturbation method

1984 ◽  
Vol 16 (1) ◽  
pp. 16-16
Author(s):  
Domokos Vermes
Keyword(s):  
Bellman Equation ◽  
Value Function ◽  
Sufficient Optimality Condition ◽  
Hamilton Jacobi Bellman Equation ◽  
Deterministic Processes ◽  
Random Jumps ◽  
Hamilton Jacobi Bellman ◽  
Necessary And Sufficient ◽  
The Value Function ◽  
Deterministic Trajectories

We consider the optimal control of deterministic processes with countably many (non-accumulating) random iumps. A necessary and sufficient optimality condition can be given in the form of a Hamilton-jacobi-Bellman equation which is a functionaldifferential equation with boundary conditions in the case considered. Its solution, the value function, is continuously differentiable along the deterministic trajectories if. the random jumps only are controllable and it can be represented as a supremum of smooth subsolutions in the general case, i.e. when both the deterministic motion and the random jumps are controlled (cf. the survey by M. H. A. Davis (p.14)).

Necessary and sufficient conditions for optimal controls in viscous flow problems

1994 ◽  
Vol 124 (2) ◽  
pp. 211-251 ◽  
Author(s):  
H. O. Fattorini ◽  
S. S. Sritharan
Keyword(s):  
Maximum Principle ◽  
Viscous Flow ◽  
Sufficient Conditions ◽  
Optimal Controls ◽  
The Maximum Principle ◽  
Flow Problems ◽  
Hamilton Jacobi Bellman Equation ◽  
Hamilton Jacobi Bellman ◽  
Necessary And Sufficient ◽  
The Pontryagin Maximum Principle

A class of optimal control problems in viscous flow is studied. Main results are the Pontryagin maximum principle and the verification theorem for the Hamilton–Jacobi–Bellman equation characterising the feedback problem. The maximum principle is established by two quite different methods.

scholarly journals THE MAIN HAMILTON FUNCTION AND THE NECESSARY CONDITIONS FOR THE EXISTENCE OF A SITUATION IN THE FORM OF THE HAMILTON-JACOBI EQUATIONS

2021 ◽  
Vol 73 (1) ◽  
pp. 32-41
Author(s):  
Sh.D. Makhmudova ◽  
◽  
A.N. Urazgalieva ◽  
Keyword(s):  
Learning Outcomes ◽  
Differential Games ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Analytical Mechanics ◽  
Equilibrium Conditions ◽  
Holistic Understanding ◽  
Theory Of Optimal Control ◽  
Necessary And Sufficient ◽  
Jacobi Equations

Interdisciplinary application of learning outcomes is essential to the field of mathematics, attainable at intersection of math with other subjects – which includes applied tasks When studying a number of technical disciplines, as well as solving applied problems, it is possible to use certain aspects of the theory of optimal control - which is an example of interdisciplinary link. Analytical mechanics, among other disciplines, enables leverage of certain aspects of the theory of differential games, namely, equilibrium conditions in non-coalitional differential games of several players. This article provides studies of the necessary and sufficient conditions for the existence of equilibrium situations, using some concepts of analytical mechanics. In line with Hamilton’s definition, necessary conditions were obtained in the form of Hamilton-Jacobi equations. This form of necessary conditions in differential games of N persons is of interest to students of natural and technical fields. The main goal of the article is to demonstrate interdisciplinary link, an important component of the process of training future engineers for various sectors of the economy. It is necessary for the holistic understanding of the material, so that students of technical specialties of various fields can use it. Proposed work can aid in study of this area of analytical mechanics by university students and young scientists alike.

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