Unique Saddle Point Research Articles

The large N expansion in hyperbolic sigma models

Invariant correlation functions for SO(1,N) hyperbolic sigma models are investigated. The existence of a large N asymptotic expansion is proven on finite lattices of dimension d≥2. The unique saddle point configuration is characterized by a negative gap vanishing at least like 1/V with the volume. Technical difficulties compared to the compact case are bypassed using horospherical coordinates and the matrix-tree theorem.

Endogenous transportation technology in a Cournot differential game with intraindustry trade

We investigate a dynamic Cournot duopoly with intraindustry trade, where firms invest in R&D to reduce the level of iceberg transportation costs. We adopt both open-loop and closed-loop equilibrium concepts, showing that a unique (saddle point) steady state exists in both cases. In the open-loop model, optimal investments and the resulting efficiency of transportation technology are independent of the relative size of the two countries. On the contrary, in the closed-loop case firms’ R&D incentives are driven by the relative size of the two countries. Policy implications are also evaluated.

Persuasive advertising under Bertrand competition: A differential game

We investigate a linear state differential game of advertising, under Cournot and Bertrand competition. A unique saddlepoint equilibrium exists if the marginal cost of advertising is sufficiently low. Bertrand competition entails more intense advertising than Cournot competition, since increasing market size is more important to firms when competition is tough.

Policy Iterations on the Hamilton–Jacobi–Isaacs Equation for $H_{\infty}$ State Feedback Control With Input Saturation

An Hinfin suboptimal state feedback controller for constrained input systems is derived using the Hamilton-Jacobi-Isaacs (HJI) equation of a corresponding zero-sum game that uses a special quasi-norm to encode the constraints on the input. The unique saddle point in feedback strategy form is derived. Using policy iterations on both players, the HJI equation is broken into a sequence of differential equations linear in the cost for which closed-form solutions are easier to obtain. Policy iterations on the disturbance are shown to converge to the available storage function of the associated L2-gain dissipative dynamics. The resulting constrained optimal control feedback strategy has the largest domain of validity within which L2-performance for a given gamma is guaranteed

Correlated Jamming on MIMO Gaussian Fading Channels

We consider a zero-sum mutual information game on multiple-input multiple-output (MIMO) Gaussian Rayleigh-fading channels. The players are an encoder-decoder pair as the maximizer, and a jammer as the minimizer, of the mutual information between the input and the output of the channel. There are total power constraints on both the jammer and the encoder. Also, the jammer has access to the encoder output. We find the unique saddle point of this game, and prove the somewhat surprising result that the knowledge of the channel input is useless to the jammer.

Effect of a complex formation on the calculated low-pressure rate constant of a bimolecular gas-phase reaction governed by tunneling

Many important bimolecular hydrogen-transfer processes that take place in the atmosphere proceed via a potential energy minimum (hydrogen-bonded complex) that precedes along the minimum energy path the unique saddle point of the reaction, the one corresponding to the hydrogen transfer. It is clear that the one-step low-pressure rate constant of such a reaction does not depend on the existence of any complex along the minimum energy path below the reactant if the reaction takes place by thermal activation over a transition state that lies quite above the reactants (for instance 10 kcal/mol). However, we have quantitatively shown in this article that the scenario notoriously changes if the reaction involves significant tunneling. In this work, we have theoretically calculated the rate constants and their temperature dependence for the reaction HO+HOH→HOH+OH by means of a canonical variational transition state theory and a canonical unified statistical theory (when necessary). Multidimensional tunneling effects have been included with a semiclassical transmission coefficient. Two kinds of modified potential energy surfaces (PESs), obtained from an original ab initio potential energy surface, previously calculated by us, have been used. The Eckart-modified PESs serve to model the hydrogen-abstraction profiles with no complexes along the path, while the Gaussian-modified PESs model the energy profiles with two complexes along the path symmetrically distributed at each side of the abstraction saddle point. Our results show that the existence of those complexes reduces the thickness of the classically forbidden region for energies below the adiabatic barrier, and then tunneling is promoted and the reaction is accelerated. The effect of the complex formation in several kinetic magnitudes, as the Arrhenius parameters and the kinetic isotope effect has also been analyzed. © 1999 John Wiley & Sons, Inc. J Comput Chem 20: 1685–1692, 1999

Dynamics of a Periodic Differential Equation with a Singular Nonlinearity of Attractive Type

where c G 0, a ) 0, and p is a continuous T-periodic function for some T ) 0. We are interested in the existence and stability of positive TŽ . periodic solutions of 1 . The existence of T-periodic solutions of this class of equations where the restoring force is a singular nonlinearity that becomes infinite in zero w x has been proved by Lazer and Solimini 2 for the case without friction Ž . w x c s 0 and by Habets and Sanchez 1 for the damped case. However, the stability properties of these solutions have been less studied. It is well Ž Ž . . known that for the autonomous case p t ' p ) 0 , this equation has a 0 unique saddle point. In this paper we prove that the dynamics of the Ž . periodic equation 1 is similar to the autonomous case. For our study is n w x fundamental a Massera’s convergence theorem in R given by Smith in 6 . The paper is divided in three sections. In Section 1, the main results are Ž . stated. It is seen that 1 has a unique unstable periodic solution w.

Singular linear quadratic differential games with bounds on control

A class of two-persons zero-sum differential games with the possibility of singular arcs is considered. Under some assumptions, a sufficient condition for the existence of a saddle-point solution is derived. In this approach, the properties of convexity and concavity of the performance index are used to find the sufficient condition. It is observed that, unlike total singularity, the occurrence of partially singular arcs does not cause any problems for the existence of a unique saddle-point solution.

Saddle Points and Ritz–Galerkin Approximations

This paper establishes a theorem for Ritz–Galerkin approximations to a unique saddle point of a function defined on a reflexive Banach space. The functional value at the saddle point is given in terms of an infimum and supremum over two closed, possibly infinite dimensional subspaces whose direct sum determines the original Banach space. Also, an application is given to a nonlinear boundary value problem involving the biharmonic operator.

Linear Quadratic Differential Games in a Hilbert Space

We consider linear quadratic games in a Hilbert space. The system equation is linear and involves an unbounded operator which generates a strongly continuous evolution operator (or semigroup). We show that the existence of a solution to a Riccati integral equation implies the existence of a saddle point for the closed-loop game, and that the former is guaranteed if there exists a unique open-loop saddle point. We also consider quadratic games on an infinite interval.

Existence of optimal open-loop strategies for a class of differential games

In this paper, the class of differential games with linear system equations and a quadratic performance index is investigated for saddlepoint solutions when one or both of the players use open-loop control. For each formulation of the game, a necessary and sufficient condition is obtained for the existence of an optimal strategy pair that generates a regular optimal path. For those cases where a solution exists, the unique saddle-point solution is presented. Also, relationships are established between the time intervals of existence of solutions for the various formulations of the game.