2,634 research outputs found
Linear-Quadratic -person and Mean-Field Games with Ergodic Cost
We consider stochastic differential games with players, linear-Gaussian
dynamics in arbitrary state-space dimension, and long-time-average cost with
quadratic running cost. Admissible controls are feedbacks for which the system
is ergodic. We first study the existence of affine Nash equilibria by means of
an associated system of Hamilton-Jacobi-Bellman and
Kolmogorov-Fokker-Planck partial differential equations. We give necessary and
sufficient conditions for the existence and uniqueness of quadratic-Gaussian
solutions in terms of the solvability of suitable algebraic Riccati and
Sylvester equations. Under a symmetry condition on the running costs and for
nearly identical players we study the large population limit, tending to
infinity, and find a unique quadratic-Gaussian solution of the pair of Mean
Field Game HJB-KFP equations. Examples of explicit solutions are given, in
particular for consensus problems.Comment: 31 page
On the relationship between the open-loop Nash equilibrium in LQ-games and the inertia of a matrix
In this paper we consider the location of the eigenvalues of the composite matrix ( -A S1 S2 ) ( Q1 At 0 ) ( Q2 0 At ) , where the matrices Si and Qi are assumed to be semi-positive definite. Two interesting observations, which are not or only partially mentioned in literature before, challenge this study. The first observation is that this matrix appears naturally in a both necessary and sufficient condition for the existence of a unique open-loop Nash solution in the 2-player linear-quadratic dynamic game and, more in particular, its inertia play an important role in the analysis of the convergence of the associated state in this game. The second observation is that from the eigenvalue and eigenstructure of this matrix all solutions for the algebraic Riccati equations corresponding with the above mentioned dynamic game can be directly calculated and, moreover, also the eigenvalues of the associated closed-loop system. Simulation experiments suggest that the composite matrix will have at least n eigenvalues (here n is the state dimension of the system) with a positive real part. Unfortunately, it turns out that this property of the inertia of this matrix in general does not hold. Some specific cases for which the property does hold are discussed.Game Theory;Nash Equilibrium;game theory
Wave impedance matrices for cylindrically anisotropic radially inhomogeneous elastic solids
Impedance matrices are obtained for radially inhomogeneous structures using
the Stroh-like system of six first order differential equations for the time
harmonic displacement-traction 6-vector. Particular attention is paid to the
newly identified solid-cylinder impedance matrix appropriate
to cylinders with material at , and its limiting value at that point, the
solid-cylinder impedance matrix . We show that
is a fundamental material property depending only on the elastic moduli and the
azimuthal order , that is Hermitian and is
negative semi-definite. Explicit solutions for are presented
for monoclinic and higher material symmetry, and the special cases of and
1 are treated in detail. Two methods are proposed for finding , one based on the Frobenius series solution and the other using a
differential Riccati equation with as initial value. %in a
consistent manner as the solution of an algebraic Riccati equation. The
radiation impedance matrix is defined and shown to be non-Hermitian. These
impedance matrices enable concise and efficient formulations of dispersion
equations for wave guides, and solutions of scattering and related wave
problems in cylinders.Comment: 39 pages, 2 figure
The extended symplectic pencil and the finite-horizon LQ problem with two-sided boundary conditions
This note introduces a new analytic approach to the solution of a very
general class of finite-horizon optimal control problems formulated for
discrete-time systems. This approach provides a parametric expression for the
optimal control sequences, as well as the corresponding optimal state
trajectories, by exploiting a new decomposition of the so-called extended
symplectic pencil. Importantly, the results established in this paper hold
under assumptions that are weaker than the ones considered in the literature so
far. Indeed, this approach does not require neither the regularity of the
symplectic pencil, nor the modulus controllability of the underlying system. In
the development of the approach presented in this paper, several ancillary
results of independent interest on generalised Riccati equations and on the
eigenstructure of the extended symplectic pencil will also be presented
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