On the existence of periodic solutions of Rayleigh equations

In this paper, we study the existence of periodic solutions of Rayleigh equation

On Global Optimality Conditions for Nonlinear Optimal Control Problems

Let a trajectory and control pair maximize globally the functional g(x(T)) in the basic optimal control problem. Then (evidently) any pair (x,u) from the level set of the functional g corresponding to the value g( (T)) is also globally optimal and satisfies the Pontryagin maximum principle. It is shown that this necessary condition for global optimality of turns out to be a sufficient one under the additional assumption of nondegeneracy of the maximum principle for every pair (x,u) from the above-mentioned level set. In particular, if the pair satisfies the Pontryagin maximum principle which is nondegenerate in the sense that for the Hamiltonian H, we have along the pair on [0,T], and if there is no another pair (x,u) such that g(x(T))=g( (T)), then is a global maximizer.     

Back-and-forth shooting method for solving two-point boundary-value problems

The paper proposes a special iterative method for a nonlinear TPBVP of the form (t)=f(t, x(t),p(t)), (t)=g(t, x(t),p(t)), subject toh(x(0),p(0))=0,e(x(T),p(T))=0. Certain stability properties of the above differential equations are taken into consideration in the method, so that the integration directions associated with these equations respectively are opposite to each other, in contrast with the conventional shooting methods. Via an embedding and a Riccati-type transformation, the TPBVP is reduced to consecutive initial-value problems of ordinary differential equations. A preliminary numerical test is given by a simple example originating in an optimal control problem.     

Invariance of asymptotic stability of perturbed linear systems on Hilbert space

The questions of stabilizability of structurally perturbed or uncertain linear systems in Hilbert space of the form are considered. The operatorA is assumed to be the infinitesimal generator of aC ₀-semigroup of contractionsT(t),t0, in a Hilbert spaceX;B is a bounded linear operator from another Hilbert spaceU toX; and {P(r),r } is a family of bounded or unbounded perturbations ofA inX, where is an arbitrary set, not necessarily carrying any topology. Sufficient conditions are presented that guarantee controllability and stabilizability of the perturbed system, given that the unperturbed system has similar properties. In particular, it is shown that, for certain classes of perturbations, weak and strong stabilizability properties are preserved for the same state feedback operator.This work was supported in part by the Natural Science and Engineering Research Council of Canada under Grant No. A7109.     

A central limit theorem for solutions of the porous medium equation

We study the large–time behavior of the second moment (energy) for the flow of a gas in a N-dimensional porous medium with initial density v₀(x) 0. The density v(x, t) satisfies the nonlinear degenerate parabolic equation v_t = v^m where m > 1 is a physical constant. Assuming that for some > 0, we prove that E(t) behaves asymptotically, as t , like the energy E_B(t) of the Barenblatt-Pattle solution B(|x|, t). This is shown by proving that E(t)/E_B(t) converges to 1 at the (optimal) rate t^{–2/(N(m-1)+2)}. A simple corollary of this result is a central limit theorem for the scaled solution E(t)^N/2v(E(t)^1/2x, t).     

Existence of Positive Periodic Solutions for the Liénard Differential Equations with Weakly Repulsive Singularity

In this paper we discuss the existence of positive T-periodic solutions for the following second order differential equation

On an asymptotically autonomous system with Tikhonov type regularizing term

We study the asymptotic behaviour of the trajectories of the second order equation ${\ddot{x}(t)+\gamma \dot{x}(t)+\nabla\phi(x(t))+\varepsilon(t)x(t)=g(t)}We study the asymptotic behaviour of the trajectories of the second order equation [(x)\ddot](t)+g[(x)\dot](t)+?f(x(t))+e(t)x(t)=g(t){\ddot{x}(t)+\gamma \dot{x}(t)+\nabla\phi(x(t))+\varepsilon(t)x(t)=g(t)} , where γ > 0, g ? L¹([0,+￥[;H){g \in L^1([0,+\infty[;H)}, Φ is a C ² convex function and e{\varepsilon} is a positive nonincreasing function.     

On the existence of value in the Varaiya-Lin sense in differential games of pursuit and evasion

This paper is strictly related to Ref. 1. A pursuit-evasion game described in part by the system and is considered. The state variablesx andy are restricted, in the sense that (x(t),t) N ₁ and (y(t),t) N ₂. The existence of a value in the sense of Varaiya and Lin is proved under the assumption that the sets of all admissible trajectories for the two players are compact and the lower value is not greater than the upper value.     

A minimax optimal control problem

A control system x=f(t,x,u) is considered, and a cost functional ess supT ₀tT ₁ G(t, x(t),u(t)) is to be minimized. Necessary conditions for optimality (maximum principle and transversality conditions) are derived. It is also shown that an optimal control is optimal for the corresponding problem on a subinterval of [T ₀,T ₁], if a certain controllability condition is satisfied.     

Nonconvex second-order differential inclusions with memory

We prove several existence theorems for the second-order differential inclusion of the form in the case whenF or bothG andF are maps with nonconvex values in an Euclidean or Hilbert space andF(t, T(t)x) is a memory term ([T(t)x]()=x(t+)).     

Inertial Gradient-Like Dynamical System Controlled by a Stabilizing Term

Let H be a real Hilbert space and let be a function that we wish to minimize. For any potential and any control function which tends to zero as t+, we study the asymptotic behavior of the trajectories of the following dissipative system:

The calculation of Stieltjes' integral

A method is proposed for the evaluation of integrals of the type f( ^p ) (x)g( ^q ) (x)dx in terms of function values off(x) andg (x). The method is based on extrapolation and is very similar to Romberg Integration. Some of the properties of the method, including its ultimate convergence, are discussed.Work performed under the auspices of the U. S. Atomic Energy Commission.     

On controllability of systems of the form\dot x=A(t)x+g(t,u)

In an earlier paper, the author established a sufficient condition for controllability of systems of the form =A(t)x+g(t, u). This condition is a growth condition which generalizes the concept of an asymptotically proper system introduced by LaSalle for linear systems. The purpose of this paper is examine and apply this growth condition. We first show that the condition is also necessary for controllability. Then, we use these results to consider the controllability of perturbations of the above system. The main result of the paper is a class of systems which in many applications can be assumed to be controllable.During the writing of this paper, the author held a Junior Faculty Summer Fellowship from the Research Council of the University of Nebraska.     

Global dynamics of delay differential equations

In this survey paper the delay differential equation (t) = −μx(t) + g(x(t − 1)) is considered with μ ≥ 0 and a smooth real function g satisfying g(0) = 0. It is shown that the dynamics generated by this simple-looking equation can be very rich. The dynamics is completely understood only for a small class of nonlinearities. Open problems are formulated. Supported in part by the Hungarian NFSR, Grant No. T049516.     

A Strongly Semismooth Integral Function and Its Application

As shown by an example, the integral function f : ⁿ , defined by f(x) = _a ^b[B(x, t)]₊ g(t) dt, may not be a strongly semismooth function, even if g(t) 1 and B is a quadratic polynomial with respect to t and infinitely many times smooth with respect to x. We show that f is a strongly semismooth function if g is continuous and B is affine with respect to t and strongly semismooth with respect to x, i.e., B(x, t) = u(x)t + v(x), where u and v are two strongly semismooth functions in ⁿ . We also show that f is not a piecewise smooth function if u and v are two linearly independent linear functions, g is continuous and g 0 in [a, b], and n 2. We apply the first result to the edge convex minimum norm network interpolation problem, which is a two-dimensional interpolation problem.     

The numerical solution of boundary value problems for second order functional differential equations by finite differences

The boundary value problem , 0 <t < 1,x(0)=x(1)=0, is considered. Hereg:R ²R ¹ andF:C[0, 1] C[0, 1]. The solutionx is approximated using finite differences. For a large class of problems it is proved that the approximate solutions exist and converge tox. The method is illustrated by the numerical example.Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No.: DA-31-124-ARO-D-462, and the Office of Naval Research under Contract No.: N00014-67-A-0128-0004. The computations were supported by the University of Wisconsin Grants Committee.     

General technique for solving nonlinear,two-point boundary-value problems via the method of particular solutions

In this paper, a general technique for solving nonlinear, two-point boundary-value problems is presented; it is assumed that the differential system has ordern and is subject top initial conditions andq final conditions, wherep+q=n. First, the differential equations and the boundary conditions are linearized about a nominal functionx(t) satisfying thep initial conditions. Next, the linearized system is imbedded into a more general system by means of a scaling factor , 01, applied to each forcing term. Then, themethod of particular solutions is employed in order to obtain the perturbation x(t)=A(t) leading from the nominal functionx(t) to the varied function (t); this method differs from the adjoint method and the complementary function method in that it employs only one differential system, namely, the nonhomogeneous, linearized system.The scaling factor (or stepsize) is determined by a one-dimensional search starting from =1 so as to ensure the decrease of the performance indexP (the cumulative error in the differential equations and the boundary conditions). It is shown that the performance index has a descent property; therefore, if is sufficiently small, it is guaranteed that <P. Convergence to the desired solution is achieved when the inequalityP is met, where is a small, preselected number.In the present technique, the entire functionx(t) is updated according to (t)=x(t)+A(t). This updating procedure is called Scheme (a). For comparison purposes, an alternate procedure, called Scheme (b), is considered: the initial pointx(0) is updated according to (0)=x(0)+A(0), and the new nominal function (t) is obtained by forward integration of the nonlinear differential system. In this connection, five numerical examples are presented; they illustrate (i) the simplicity as well as the rapidity of convergence of the algorithm, (ii) the importance of stepsize control, and (iii) the desirability of updating the functionx(t) according to Scheme (a) rather than Scheme (b).This research, supported by the National Science Foundation, Grant No. GP-18522, is based on Ref. 1. The authors are indebted to Mr. A. V. Levy for computational assistance.     

An Example on the Controllability of Perturbed Linear Control Systems

We show by an example that small perturbations with respect to the convergence in measure of the coefficients A and B may affect the complete controllability of the linear control system , where x is the state and u is the control. This answers a question raised in Ref. 1.     

Controllability of Nonlinear Evolution Systems with Preassigned Responses

In this paper, we study the approximate controllability with preassigned responses of the nonlinear delay systems x(t)=A(t)x(t)+f(t, x(t), x((t)), u(t)) and L(x(t), x(t))=A(t)x(t)+f(t, x(t), x((t)), u(t)). The controllability is not governed by an associated linear system, but by conditions on f or A involving the domain of A(t). No compactness assumptions are imposed in the main results.     

Steepest Descent with Curvature Dynamical System

Let H be a real Hilbert space and let <..,.> denote the corresponding scalar product. Given a function that is bounded from below, we consider the following dynamical system: