A New Approach to Linearly Perturbed Riccati Equations Arising in Stochastic Control

In this paper a linearly perturbed version of the well-known matrix Riccati equations which arise in certain stochastic optimal control problems is studied. Via the concepts of mean square stabilizability and mean square detectability we improve previous results on both the convergence properties of the linearly perturbed Riccati differential equation and the solutions of the linearly perturbed algebraic Riccati equation. Furthermore, our approach unifies, in some way, the study for this class of Riccati equations with the one for classical theory, by eliminating a certain inconvenient assumption used in previous works (e.g., [10] and [26]). The results are derived under relatively weaker assumptions and include, inter alia, the following: (a) An extension of Theorem 4.1 of [26] to handle systems not necessarily observable. (b) The existence of a strong solution, subject only to the mean square stabilizability assumption. (c) Conditions for the existence and uniqueness of stabilizing solutions for systems not necessarily detectable. (d) Conditions for the existence and uniqueness of mean square stabilizing solutions instead of just stabilizing. (e) Relaxing the assumptions for convergence of the solution of the linearly perturbed Riccati differential equation and deriving new convergence results for systems not necessarily observable. Accepted 30 July 1996     

Nonautonomous regulator problem in Hilbert spaces

In this paper, we study the quadratic optimal control problem on the half linett _o, for nonautonomous control processes in Hilbert spaces. We prove that the quadratic optimal control problem has a solution if, and only if, an associated Riccati equation has a positive solution fortt _o. The optimal control is given in feedback form. If a detectability assumption holds, then we prove that the optimal control is a stabilizing feedback control when the associated Riccati equation has a positive solution which is bounded fortt _o.This work was performed under the auspices of the National Research Council of Italy (CNR).     

Reid Roundabout Theorem for Symplectic Dynamic Systems on Time Scales

The principal aim of this paper is to state and prove the so-called Reid roundabout theorem for the symplectic dynamic system (S) z ^Δ = \cal S _t z on an arbitrary time scale \Bbb T , so that the well known case of differential linear Hamiltonian systems ( \Bbb T = \Bbb R ) and the recently developed case of discrete symplectic systems ( \Bbb T = \Bbb Z ) are unified. We list conditions which are equivalent to the positivity of the quadratic functional associated with (S), e.g. disconjugacy (in terms of no focal points of a conjoined basis) of (S), no generalized zeros for vector solutions of (S), and the existence of a solution to the corresponding Riccati matrix equation. A certain normality assumption is employed. The result requires treatment of the quadratic functionals both with general and separated boundary conditions. Accepted 28 August 2000. Online publication 26 February 2001.     

Adaptive control of linear stochastic evolution systems

An adaptive control problem for some linear stochastic evolution systems in Hilbert spaces is formulated and solved in this paper. The solution includes showing the strong consistency of a family of least squares estimates of the unknown parameters and the convergence of the average quadratic costs with a control based on these estimates to the optimal average cost. The unknown parameters in the model appear affinely in the infinitesimal generator of the C ₀ semigroup that defines the evolution system. A recursive equation is given for a family of least squares estimates and the bounded linear operator solution of the stationary Riccati equation is shown to be a continuous function of the unknown parameters in the uniform operator topology     

Computation of the stabilizing solution of game theoretic Riccati equation arising in stochastic <Emphasis Type="Italic">H</Emphasis><Subscript> ∞ </Subscript> control problems

In this paper, the problem of the numerical computation of the stabilizing solution of the game theoretic algebraic Riccati equation is investigated. The Riccati equation under consideration occurs in connection with the solution of the H _∞ control problem for a class of stochastic systems affected by state dependent and control dependent white noise. The stabilizing solution of the considered game theoretic Riccati equation is obtained as a limit of a sequence of approximations constructed based on stabilizing solutions of a sequence of algebraic Riccati equations of stochastic control with definite sign of the quadratic part. The efficiency of the proposed algorithm is demonstrated by several numerical experiments.     

LARGE TIME BEHAVIOR OF SOLUTIONS TO SOME DEGENERATE PARABOLIC EQUATIONS

The purpose of this paper is to study the limit in L ¹(Ω), as t → ∞, of solutions of initial-boundary-value problems of the form u_t ? Δw = 0 and u ∈ β(w) in a bounded domain Ω with general boundary conditions ?w/?η + γ(w) ? 0. We prove that a solution stabilizes by converging as t → ∞ to a solution of the associated stationary problem. On the other hand, since in general these solutions are not unique, we characterize the true value of the limit and comment the results on the related concrete situations like the Stefan problem and the filtration equation.     

On the Behavior of a Nonstationary Poiseuille Solution as <Emphasis Type="Italic">t</Emphasis> → ∞

A nonstationary Poiseuille solution describing the flow of a viscous incompressible fluid in an infinite cylinder is defined as a solution to an inverse problem for the heat equation. The behavior as t → ∞ of the nonstationary Poiseuille solution corresponding to the prescribed flux F(t) ofthe velocity field is studied. In particular, it is proved that if the flux F(t) tends exponentially to a constant flux F _* then the nonstationary Poiseuille solution tends exponentially as t → ∞ to the stationary Poiseuille solution having the flux F _*.Original Russian Text Copyright © 2005 Pileckas K.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 890–900, July–August, 2005.     

Convergence of the solution of a nonsymmetric matrix Riccati differential equation to its stable equilibrium solution

We consider the initial value problem for a nonsymmetric matrix Riccati differential equation, where the four coefficient matrices form an M-matrix. We show that for a wide range of initial values the Riccati differential equation has a global solution X(t) on [0,∞) and X(t) converges to the stable equilibrium solution as t goes to infinity.     

Uniqueness theorems for a nonlinear Tricomi problem and the related evolution problem

We consider an initial-boundary value problem for the non-linear evolution equation in a cylinder Q_t = Ω × (0, t), where T[u] = yu_xx + u_yy is the Tricomi operator and l(u) a special differential operator of first order. In [10] we proved the existence of a generalized solution of problem (1) and the existence of a generalized solution of the corresponding stationary boundary value problem (non-linear Tricomi problem) In this paper we give sufficient conditions for the uniqueness of these solutions.     

Sensitivity analysis of the differential matrix Riccati equation based on the associated linear differential system

Perturbation bounds are given for the solution of the nth order differential matrix Riccati equation using the associated linear 2nth order differential system. The new bounds are alternative to those existing in the literature and are sharper in some cases. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the stability and domain of attraction of asymptotically nonsmooth stationary solutions to a singularly perturbed parabolic equation

A stationary solution to the singularly perturbed parabolic equation ?u _t + ε² u _xx ? f(u, x) = 0 with Neumann boundary conditions is considered. The limit of the solution as ε → 0 is a nonsmooth solution to the reduced equation f(u, x) = 0 that is composed of two intersecting roots of this equation. It is proved that the stationary solution is asymptotically stable, and its global domain of attraction is found.     

Exact slow-fast decomposition of the singularly perturbed matrix differential Riccati equation

In this paper the Hamiltonian matrix formulation of the Riccati equation is used to derive the reduced-order pure-slow and pure-fast matrix differential Riccati equations of singularly perturbed systems. These pure-slow and pure-fast matrix differential Riccati equations are obtained by decoupling the singularly perturbed matrix differential Riccati equation of dimension n₁+n₂ into the pure-slow regular matrix differential Riccati equation of dimension n₁ and the pure-fast stiff matrix differential Riccati equation of dimension n₂. A formula is derived that produces the solution of the original singularly perturbed matrix differential Riccati equation in terms of solutions of the pure-slow and pure-fast reduced-order matrix differential Riccati equations and solutions of two reduced-order initial value problems. In addition to its theoretical importance, the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear time-invariant systems independently in pure-slow and pure-fast time scales.     

From the Ehrenfest model to time-fractional stochastic processes

The Ehrenfest model is considered as a good example of a Markov chain. I prove in this paper that the time-fractional diffusion process with drift towards the origin, is a natural generalization of the modified Ehrenfest model. The corresponding equation of evolution is a linear partial pseudo-differential equation with fractional derivatives in time, the orders lying between 0 and 1. I focus on finding a precise explicit analytical solution to this equation depending on the interval of the time. The stationary solution of this model is also analytically and numerically calculated. Then I prove that the difference between the discrete approximate solution at time t_n, n≥0, and the stationary solution obeys a power law with exponent between 0 and 1. The reversibility property is discussed for the Ehrenfest model and its fractional version with a new observation.     

On the Feynman-Kac formula and its applications to filtering theory

In this article the Feynman-Kac formula is obtained for a Markov process (X _t) whose transition probability function is not stationary. A converse to the Feynman-Kac formula is also obtained. This is used to prove the uniqueness of the solution to a measure-valued equation satisfied by the optimal filter in the white-noise approach to nonlinear filtering theory.Research partially supported by the Air Force Office of Scientific Research Contract No. F49620 85 C 0144 and by the Indian Statistical Institute.     

Normal forms for stochastic differential equations

Summary.   We address the following problem from the intersection of dynamical systems and stochastic analysis: Two SDE dx _t = ∑ _{j =0} ^m f _j (x _t )∘dW _t ^j and dx _t =∑ _{j =0} ^m g _j (x _t )∘dW _t ^j in ℝ ^d with smooth coefficients satisfying f _j (0)=g _j (0)=0 are said to be smoothly equivalent if there is a smooth random diffeomorphism (coordinate transformation) h(ω) with h(ω,0)=0 and Dh(ω,0)=id which conjugates the corresponding local flows,

On the Logistic Equation in the Complex Plane

The famous logistic differential equation is studied in the complex plane. The method used is based on a functional analytic technique which provides a unique solution of the ordinary differential equation (ODE) under consideration in H ₂(𝔻) or H ₁(𝔻) and gives rise to an equivalent difference equation for which a unique solution is established in ?₂ or ?₁. For the derivation of the solution of the logistic differential equation this discrete equivalent equation is used. The obtained solution is analytic in {z ∈ ?: |z| <T}, T > 0. Numerical experiments were also performed using the classical 4th order Runge–Kutta method. The obtained results were compared for real solutions as well as for solutions of the form y(t) = u(t) + iv(t), t ∈ ?. For t ∈ ? the solution derived by the present method, seems to have singularities, that is, points where it ceases to be analytic, in certain sectors of the complex plane. These sectors, depending on the values of the involved parameters, can move at different directions, join forming common sectors, or pass through each other and continue moving independently. Moreover, the real and imaginary part of the solution seem to exhibit oscillatory behavior near these sectors.     

Filtering on manifolds

We coasider a partially observable diffusion process (x _t,y_t)t0 whose unobservable componentx _t lives on a submanifold M ofR ⁿ . We present some general conditions under which the conditional law ofx _t, given the observationsy _s ,s [0,t], admits a density w.r.t. a given measure on M. We characterize the analytical properties of this density by using appropriate Sobolev spaces.Research supported by the Hungarian National Foundation of Scientific Research No. 2290.     

Analysis of a Free Boundary Problem Modeling Tumor Growth

In this paper, we study a free boundary problem arising from the modeling of tumor growth. The problem comprises two unknown functions： R = R（t）, the radius of the tumor, and u = u（r, t）, the concentration of nutrient in the tumor. The function u satisfies a nonlinear reaction diffusion equation in the region 0 〈 r 〈 R（t）, t 〉 0, and the function R satisfies a nonlinear integrodifferential equation containing u. Under some general conditions, we establish global existence of transient solutions, unique existence of a stationary solution, and convergence of transient solutions toward the stationary solution as t →∞.     

The Perturbed Plane-Wave Solutions of the Cubic Schrödinger Equation

A detailed analysis is given to the solution of the cubic Schrödinger equation iq_t + q_xx + 2|q|²q = 0 under the boundary conditions as |x|→∞. The inverse-scattering technique is used, and the asymptotic state is a series of solitons. However, there is no soliton whose amplitude is stationary in time. Each soliton has a definite velocity and “pulsates” in time with a definite period. The interaction of two solitons is considered, and a possible extension to the perturbed periodic wave [q(x + T,t) = q(x,t) as |x|→∞] is discussed.     

Spectral theory and L∞-time decay estimates for Klein–Gordon equations on two half axes with transmission: The tunnel effect

Consider two copies N₁, N₂ of the interval [0, ∞]. Consider Klein-Gordon equations with (different) constant coefficients on ? × N_j ( = time × space). Assume the coincidence of the values of the solution at the boundary points of the N_j for all times and a transmission condition relating its first (one-sided) space derivatives at these points. Under a symmetry condition, we extend the spatial part of the equation and the transmission conditions to a self-adjoint operator (by Friedrichs extension) and reformulate our problem in terms of an abstract wave equation in a suitable Hilbert space. We derive an expansion of the solution in generalized eigenfunctions of this self-adjoint extension and show, that the L^∞-norms (in space) of the solution and its first k space derivatives at the time t decay for t → ∞ at least as const. t^¼, if the initial conditions satisfy a compatibility condition of order k derived in this paper. The loss of decay rate in comparison with the full line case (const. t^?½, cf. [28]) is caused by the tunnel effect. Further we show that an abstract wave equation in a Hilbert space with a Friedrichs extension as spatial part can always be derived from a stationarity principle for an associated action-type functional. This yields a physical legitimation of our model by the principle of stationary action and moreover a criterion for the physical interpretability of all models created by the linear interaction concept [4, 6, 8, 10], in particular for the coupling of media of different dimension (alternative to [13, 16] for similar models).