IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

We revisit some concepts of superiority and weak stability of population states in evolutionary games with continuous strategy space. We prove a general stability result for replicator trajectories by introducing the concept of superiority with respect to a given closed set. Some important results in the literature regarding weak stability turn out to be special cases of our main result. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

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Department of Mathematics

Annals of Operations Research

On superiority and weak stability of population states in evolutionary games

The concept of a globally strong uninvadable set of profiles is studied for asymmetric evolutionary games. After establishing a general result concerning weak ω-limit points of the replicator dynamics trajectories, we prove that globally strong uninvadable sets are weak attracting. © 2020 World Scientific Publishing Company.

International Game Theory Review

Globally Strong Uninvadable Sets of Profiles in Asymmetric Games

In the present study, we develop an energetically efficient suboptimal open-loop strategy to control the wake behind a circular cylinder in the laminar regime. The open-loop suboptimal controller is designed to resemble the feedback integral controller with reference to its dynamical behaviour. Energetic efficiency is measured using the power loss coefficient. The Van der Pol model for the evolution of lift force on the cylinder is chosen as the reduced-order model for the development of an open-loop suboptimal controller. The parameter estimation of the low- dimensional model is carried out using the results from the continuum based Navier - Stokes simulations. It is shown that a subspace identification method can be used to model the relationship between the inputs to the reduced-order model and the inputs to the higher-order computational fluid dynamic model. The development of the suboptimal control is realised by means of solving suitably formulated optimal tracking and regulator problems using the Pontryagin's minimum principle. The resultant controller is found to be energetically efficient and also successful in the control of vortex shedding. © 2018 Elsevier Ltd

GENERAL

Department of Applied Mechanics

Computers and Fluids

Suboptimal control of circular cylinder wakes using Van der Pol oscillator

Mendoza-Palacios and Hernández-Lerma (J Differ Equ 259(11):5709–5733, 2015) have introduced the concept of a strong uninvadable profile for asymmetric games with continuous pure strategy space and proved that such a profile is Lyapunov stable for the associated replicator dynamics when the profile is monomorphic. In the present paper, we establish that a polymorphic strong uninvadable profile is necessarily monomorphic. Further, it is shown that strong unbeatability is enough to guarantee Lyapunov stability of polymorphic profiles. A stability theorem for sets of polymorphic profiles is also presented and is illustrated using examples. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

Dynamic Games and Applications

Evolutionary Stability of Polymorphic Profiles in Asymmetric 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. © 2016, Springer Science+Business Media New York.

Fulltext

Evolutionary Stability of Polymorphic Population States in Continuous Games

The control of vortex shedding behind a circular cylinder is a precursor to a wide range of external shear flow problems in engineering, in particular the flow-induced vibrations. In the present study, numerical simulation of an energetically efficient active flow control strategy is proposed, for the control of wake vortices behind a circular cylinder at a low Reynolds number of 100. The fluid is assumed to be incompressible and Newtonian with negligible variation in properties. Reflectionally symmetric controllers are designed such that, they are located on a small sector of the cylinder over which, tangential sliding mode control is imparted. In the field of modern controls, proportional (P), integral (I) and differential (D) control strategies and their numerous combinations are extremely popular in industrial practice. To impart suitable control actuation, the vertically varying lift force on the circular cylinder, is synthesised for the construction of an error term. Four different types of controllers considered in the present study are, P, I, PI and PID. These controllers are evaluated for their energetic efficiency and performance. A linear quadratic optimal control problem is formulated, to minimise the cost functional. By performing detailed simulations, it was observed that, the system is energetically efficient, even when the twin eddies are still persisting behind the circular cylinder. To assess the adaptability of the controllers, the actuators were switched on and off to study their dynamic response. © 2016 The Japan Society of Fluid Mechanics and IOP Publishing Ltd.

Fluid Dynamics Research

Energetically efficient proportional-integral-differential (PID) control of wake vortices behind a circular cylinder

We consider a dimorphic population state, P, which is a convex combination of two Dirac measures δx and δy, 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= { ηδx+ (1 − η) δy: 0 &lt; η&lt; 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. © Springer International Publishing Switzerland 2016.

Annals of the International Society of Dynamic Games

Evolutionary stability of dimorphic population states

For affine control problems, it is known that the optimal controller for these classes of problems can be expressed in terms of the associated co-state. A method to obtain the costate vector in terms of the state vector is given in this article. Using this method, the optimal control can be written as a function of state variable. Further the method is illustrated using examples. © IFAC.

IFAC Proceedings Volumes (IFAC-PapersOnline)

Solution of affine quadratic control problems

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. © 2013 World Scientific Publishing Company.

Some remarks on evolutionary stability in matrix games

Summary This paper presents an approach to discrete-time robust H ∞ control for a class of nonlinear uncertain systems on the basis of the use of Sum Quadratic Constraints. The approach involves controllers, which include copies of the system nonlinearities in the controller. The nonlinearities being considered are those that satisfy a certain global Lipschitz condition. The linear part of the controller is synthesized using linear robust H∞ control theory, and this leads to a nonlinear controller, which gives an upper bound on the attainable disturbance attenuation level. © 2012 John Wiley &amp; Sons, Ltd.

International Journal of Robust and Nonlinear Control

Discrete-time robust H∞control of a class of nonlinear uncertain systems

A recently emerging approach to the feedback control of linear quantum systems involves the use of a controller which itself is a quantum linear system. This approach to quantum feedback control, referred to as coherent quantum feedback control, has the advantage that it does not destroy quantum information, is fast, and has the potential for efficient implementation. An important issue which arises both in the synthesis of linear coherent quantum controllers and in the modeling of linear quantum systems, is the issue of physical realizability. This issue relates to the property of whether a given set of linear quantum stochastic differential equations corresponds to a physical quantum system satisfying the laws of quantum mechanics. Under suitable assumptions, the paper shows that the question of physical realizability is equivalent to a frequency domain (J,J) -unitary condition. This is important in controller synthesis since it is the transfer function matrix of the controller which determines the closed loop system behavior. © 2012 IEEE.

IEEE Transactions on Automatic Control

A frequency domain condition for the physical realizability of linear quantum systems

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. © 2012 Springer Science+Business Media, LLC.

Evolutionary Stability Against Multiple Mutations

This paper is concerned with the physical realizability of linear Quantum Stochastic Differential Equations (QSDEs). In a recent paper, James, Nurdin and Petersen gave necessary and sufficient conditions for the physical realizability of linear QSDEs with real coefficients. These conditions were algebraic conditions on the system state space matrices. In this paper, we first study general linear QSDEs with complex coefficients and explain how to relate them via unitary transformations to the class of linear QSDEs with real coefficients considered by James, Nurdin Petersen. We use this relation between linear QSDEs with real and complex coefficients to derive necessary and sufficient algebraic conditions for the physical realizability of the general complex linear QSDEs being considered. Then we prove the equivalence between these algebraic conditions and a condition that an associated linear system is (J, J′)-unitary. ©2009 IEEE.

Proceedings of the IEEE Conference on Decision and Control

On the physical realizability of general linear quantum stochastic differential equations with complex coefficients

We revisit the relationship between evolutionarily stable strategy (ESS) and related topics such as evolutionarily robust strategy (ERS) on the one hand, and stability of the replicator dynamics on the other hand, when the phenotypic set is continuous. The state of the population considered is a measure over the phenotypic set. Thus, topological considerations come into play that make the situation much more difficult than for a finite phenotypic set. As a consequence, the issue of the asymptotic stability of an ERS is not settled at this time. We give one partial new result in that direction.It has also been noticed in the literature that there is a dearth of concrete examples of mixed ESS or ERS in the literature. Actually none seems to be knownif the “kernel” of the game is continuous.We provide two such examples, one a convex combination of two Dirac measures and one family with the Lebesgue measure as an ERS.

Advances in Dynamic Games and Their Applications

Evolutionarily Robust Strategies: Two Nontrivial Examples and a Theorem

Two-person zero-sum infinite-dimensional differential games with strategies and payoff as in Berkovitz (SIAM J. Control Optim. 23: 173-196, 1985) are studied. Using Yosida type approximations of the infinitesimal generator (of the unbounded dynamics) by bounded linear operators, we prove convergence theorems for the approximate value functions. This is used to construct approximate saddle-point strategies in feedback form. © 2009 Springer Science+Business Media, LLC.

Journal of Optimization Theory and Applications

Construction of approximate saddle-point strategies for differential games in a hilbert Space

This paper is aimed at deriving an explicit formula for the optimal cost for discrete-time linear exponential-of-quadratic Gaussian (LEQG) control problems. We make direct calculations for the general case with cross terms in the cost and noise covariance matrices using an information-state approach. © 2009 Elsevier Ltd.

Automatica

A formula for the optimal cost in the general discrete-time LEQG problem

We study the H∞ Controller reduction problem for structured uncertain systems whose uncertainties are governed by Integral Quadratic Constraints. A sufficient condition for obtaining a reduced-order controller is given in terms of a system of rank constrained Linear Matrix Inequalities which can be solved using existing algorithms. In this approach, the balancing state transformation is a decision variable allowing better performance of the reduced-order controller than the case of a fixed state transformation. The main result is illustrated by two examples. ©2009 ACA.

Proceedings of 2009 7th Asian Control Conference, ASCC 2009

H∞ controller reduction for uncertain systems via rank constrained linear matrix inequalities

our objective is to determine the evolutionarily stable strategy [13] that is supposed to drive the behavior of foragers competing for a common patchily distributed resource [15]. Compared to [17], the innovation lies in the fact that random arrival times are allowed. In this second part, we add interference to the model: it implies that a “passive” Charnov-like strategy can no longer be optimal. A dynamic programming approach leads to a sequence of wars of attrition [13] with random end times. This game is solved in Appendix A. Under some conditions that prevail in our model, the solution is independent of the probability law of the horizon. As a consequence, the solution of the asynchronous foraging problem investigated here, expressed as a closed loop strategy on the number of foragers, is identical to that of the synchronous problem [17]. Finally, we discuss the biological implications such as a possible connection with the genetic variability in the susceptibility to interference observed in [22]. © 2007 Birkhäuser Boston All rights reserved.

Foraging under competition: Evolutionarily stable patch-leaving strategies with random arrival times 2. Interference competition

In this paper, we formulate and solve a guaranteed cost control problem for a class of uncertain linear stochastic quantum systems. For these quantum systems, a connection with an associated classical (non-quantum) system is first established. The theory presented is illustrated using an example from quantum optics. © 2007 Optical Society of America.

Optics InfoBase Conference Papers

Guaranteed cost linear quadratic gaussian control of uncertain linear stochastic quantum systems

In this paper, we formulate and solve a guaranteed cost control problem for a class of uncertain linear stochastic quantum systems. For these quantum systems, a connection with an associated classical (non-quantum) system is first established. Using this connection, the desired guaranteed cost results are proved. The theory presented is illustrated using an example from quantum optics. © 2007 IEEE.