K. S. Mallikarjuna Rao

Differential Games with Ergodic Payoff

We address a zero-sum differential game with ergodic payoff. We study this problem via the viscosity solutions of an associated Hamilton--Jacobi--Isaacs equation. Under certain condition, we establish the existence of a value and prove certain representation formulae.

Characterization of maximum hands-off control

Abstract Maximum hands-off control aims to maximize the length of time over which zero actuator values are applied to a system when executing specified control tasks. To tackle such problems, recent literature has investigated optimal control problems which penalize the size of the support of the control function and thereby lead to desired sparsity properties. This article gives the exact set of necessary conditions for a maximum hands-off optimal control problem using an L 0 -norm, and also provides sufficient conditions for the optimality of such controls. Numerical example illustrates that adopting an L 0 cost leads to a sparse control, whereas an L 1 -relaxation in singular problems leads to a non-sparse solution.

A probabilistic approach to second order variational inequalities with bilateral constraints

We study a class of second order variational inequalities with bilateral constraints. Under certain conditions we show the existence of aunique viscosity solution of these variational inequalities and give a stochastic representation to this solution. As an application, we study a stochastic game with stopping times and show the existence of a saddle point equilibrium.

Evolutionary Stability of Polymorphic Population States in Continuous Games

Asymptotic stability of equilibrium in evolutionary games with continuous action spaces is an important question. Existing results in the literature require that the equilibrium state be monomorphic. In this article, we address this question when the equilibrium is polymorphic. We show that any uninvadable and finitely supported state is asymptotically stable equilibrium of replicator equation.

Evolutionary Stability of Dimorphic Population States

We consider a dimorphic population state, P, which is a convex combination of two Dirac measures δ x and δ y , in evolutionary games with a continuous strategy space. We first establish necessary and sufficient conditions for this dimorphic population state, P, to be a rest point of the associated replicator dynamics. We provide sufficient conditions for the replicator dynamics trajectory to converge to P when it originates from the line \(L =\{\eta \delta _{x} + (1-\eta )\delta _{y}: 0 <\eta < 1\}\). If the trajectory emanates from a point outside L, then we derive sufficient conditions for the trajectory to converge to L in the special case where each point in L is a rest point. We have, also, obtained condition for the trajectory to stay away from the line L in the limit. Furthermore, main results are illustrated using examples.

Existence of Value in Stochastic Differential Games of Mixed Type

In this article, we address stochastic differential games of mixed type with both control and stopping times. Under standard assumptions, we show that the value of the game can be characterized as the unique viscosity solution of corresponding Hamilton-Jacobi-Isaacs (HJI) variational inequalities.

Evolutionary Stability Against Multiple Mutations

It is known (see, e.g., Weibull in Evolutionary Game Theory, MIT Press, 1995) that an evolutionarily stable strategy is not necessarily robust against multiple mutations. Precise definition and analysis of “evolutionarily stable strategy against multiple mutations” are not available in the literature. In this article, we formalize evolutionarily robustness against multiple mutations. Our main result shows that such a robust strategy is necessarily a pure strategy. Further, we study some equivalent formulations and properties of evolutionary stability against multiple mutations. In particular, we characterize completely the robustness against multiple mutations in 2×2 games.

Some Remarks On Evolutionary Stability In Matrix Games

In this article, we revisit evolutionary stability in matrix games. We provide a new direct proof to characterize a pure evolutionarily stable strategy (ESS), in games with exactly two pure strategies, as a strategy that is evolutionarily stable against multiple mutations. This direct proof yields generalizations to k × k games which explains why such a characterization is not possible in general. Furthermore, we prove other necessary/sufficient conditions for evolutionary stability against multiple mutations.

Zero-sum stochastic games with stopping and control

We study a zero-sum stochastic game where each player uses both control and stopping times. Under certain conditions we establish the existence of a saddle point equilibrium, and show that the value function of the game is the unique solution of certain dynamic programming inequalities with bilateral constraints.

A zero sum differential game in a Hilbert space

We study a zero sum differential game of fixed duration in a separable Hilbert space. We prove a minimax principle and establish the equivalence between the dynamic programming principle and the existence of a saddle point equilibrium. We also prove sufficient conditions for optimality.