unbounded perturbation of the exponential dichotomy for evolution equations abstract in this paper we prove that the exponential dichotomy for evolution equations in banach spaces is not destroyed if we perturb the equation by small unbounded linear operator this is done by employing a skew product semiflow technique and a perturbation principle from linear operator theory finally we apply these results to partial parabolic equations and functional differential equations

fredholm differential operators with unbounded coefficients abstract we prove that a first order linear differential operator g with unbounded operator coefficients is fredholm on spaces of functions on r with values in a reflexive banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both r and r and a pair of the ranges of the dichotomy projections is fredholm and that the fredholm index of g is equal to the fredholm index of the pair the operator g is the generator of the evolution semigroup associated with the evolution family in the case when the evolution family is the propagator of a well posed differential equation u t a t u t with generally unbounded operators a t t r the operator g is a closure of the operator d dt a t thus this paper provides a complete infinite dimensional generalization of well known finite dimensional results by palmer and by ben artzi and gohberg

perturbation and an abstract characterization of evolution semigroups abstract evans and howland used an abstract characterization of evolution semigroups and a perturbation result for c 0 semigroups to treat non autonomous schrodinger equations and problems in scattering theory here we proceed in a similar way at first we characterize evolution semigroups on spaces of vector valued functions e x which are induced by a strongly continuous evolution family u t s t s defined on the banach space x from this and a perturbation result of voigt we obtain a sufficient condition for an operator on l p r x to be the generator of an evolution semigroup this will be applied to non autonomous heat equations with absorption

online solution of two player zero sum games for continuous time nonlinear systems with completely unknown dynamics regarding two player zero sum games of continuous time nonlinear systems with completely unknown dynamics this paper presents an online adaptive algorithm for learning the nash equilibrium solution i e the optimal policy pair first for known systems the simultaneous policy updating algorithm spua is reviewed a new analytical method to prove the convergence is presented then based on the spua without using a priori knowledge of any system dynamics an online algorithm is proposed to simultaneously learn in real time either the minimal nonnegative solution of the hamilton jacobi isaacs hji equation or the generalized algebraic riccati equation for linear systems as a special case along with the optimal policy pair the approximate solution to the hji equation and the admissible policy pair is reexpressed by the approximation theorem the unknown constants or weights of each are identified simultaneously by resorting to the recursive least square method the convergence of the online algorithm to the optimal solutions is provided a practical online algorithm is also developed simulation results illustrate the effectiveness of the proposed method

equivalences between nonuniform exponential dichotomy and admissibility abstract relationship between exponential dichotomies and admissibility of function classes is a significant problem for hyperbolic dynamical systems it was proved that a nonuniform exponential dichotomy implies several admissible pairs of function classes and conversely some admissible pairs were found to imply a nonuniform exponential dichotomy in this paper we find an appropriate admissible pair of classes of lyapunov bounded functions which is equivalent to the existence of nonuniform exponential dichotomy on half lines r separately on both half lines r simultaneously and on the whole line r additionally the maximal admissibility is proved in the case on both half lines r simultaneously

perturbation theory for the approximation of stability spectra by qr methods for sequences of linear operators on a hilbert space abstract in this paper we develop a perturbation analysis for stability spectra lyapunov exponents and sacker sell spectrum for products of operators on a hilbert space both real and complex based upon the discrete qr technique error bounds are obtained in both the integrally separated and non integrally separated cases that correspond to distinct and multiple eigenvalues respectively for a single linear operator we illustrate our results using a linear parabolic partial differential equation in which the strength of the integral separation the time varying analogue of gaps between eigenvalues determines the sensitivity of the stability spectra to perturbation

existence and roughness of the exponential dichotomy for skew product semiflow in banach spaces abstract in this paper we introduce a concept of exponential dichotomy for skew product semiflow in infinite dimensional banach spaces which is an extension of the classic concept for evolution operators this concept is used to study the roughness property of the skew product semiflow also we introduce the concept of discrete skew product and give a necessary and sufficient condition for this discrete skew product to have a discrete dichotomy after that we give necessary and sufficient conditions for the existence of exponential dichotomy for skew product semiflow moreover we prove that the exponential dichotomy for skew product semiflow is not destroyed by small perturbation finally we apply these results to parabolic partial differential equations and functional differential equations

an iterative algorithm to solve state perturbed stochastic algebraic riccati equations in lq zero sum games an iterative algorithm to solve a kind of state perturbed stochastic algebraic riccati equation sare in lq zero sum game problems is proposed in our algorithm we replace the problem of solving a sare with an indefinite quadratic term by the problem of solving a sequence of sares with a negative semidefinite quadratic term which can be solved by existing methods under some appropriate conditions we prove that our algorithm is globally convergent we give a numerical example to show the effectiveness of our algorithm our algorithm also has a natural game theoretic interpretation

admissibility and exponential trichotomy of dynamical systems described by skew product flows abstract the aim of this paper is to present a new and very general method for the detection of the uniform exponential trichotomy of dynamical systems the investigation is done in several constructive stages that correspond to three admissibility properties that are progressively introduced with respect to an associated input output system we prove that the uniform admissibility of the pair c b r x l 1 r x for the associated system is a sufficient condition for the existence of a uniform trichotomic behavior of the initial dynamical system if p 1 and the pair c b r x l 1 r x is uniformly p admissible then we obtain the uniform exponential trichotomy next we study whether the admissibility conditions are also necessary for the uniform exponential trichotomy supposing that a dynamical system has a uniform exponential trichotomy we prove that the associated input output system has unique bounded solutions in certain subspaces finally we obtain that the uniform p admissibility of the pair c b r x l 1 r x is a necessary and sufficient condition for uniform exponential trichotomy

robust adaptive dynamic programming of two player zero sum games for continuous time linear systems in this brief an online robust adaptive dynamic programming algorithm is proposed for two player zero sum games of continuous time unknown linear systems with matched uncertainties which are functions of system outputs and states of a completely unknown exosystem the online algorithm is developed using the policy iteration pi scheme with only one iteration loop a new analytical method is proposed for convergence proof of the pi scheme the sufficient conditions are given to guarantee globally asymptotic stability and suboptimal property of the closed loop system simulation studies are conducted to illustrate the effectiveness of the proposed method

exponential dichotomies and homoclinic orbits in functional differential equations abstract suppose an autonomous functional differential equation has an orbit which is homoclinic to a hyperbolic equilibrium point the purpose of this paper is to give a procedure for determining the behavior of the solutions near of a functional differential equation which is a nonautonomous periodic perturbation of the original one the procedure uses exponential dichotomies and the fredholm alternative it is also shown that any smooth function p t defined on the reals which approaches zero monotonically as t is the solution of a scalar functional differential equation and generates an orbit homoclinic to zero examples illustrating the results are also given

nonuniform h k μ ν dichotomy with applications to nonautonomous dynamical systems the paper develops and studies a very general notion of dichotomy referred to as nonuniform h k dichotomy the new notion contains as special cases most versions of dichotomy existing in the literature the paper then provides corresponding new versions of robustness hartman grobman theorem and stable manifold theorem for nonautonomous dynamical systems in banach spaces in term of the nonuniform h k dichotomy

heteroclinic orbits for retarded functional differential equations abstract suppose is a heteroclinic orbit of a ck functional differential equation x t x i with limit set and limit set being either hyperbolic equilibrium points or periodic orbits necessary and sufficient conditions are given for the existence of an ftf close to in ck with the property that x t tf x t has a heteroclinic orbit gg close to the orbits gg are obtained from the zeros of a finite number of bifurcation functions g f r d r d 1 transversality of is characterized by the nondegeneracy of the derivative d g it is also shown that the tf which have heteroclinic orbits near are on a ck submanifold of finite codimension max 0 ind or on the closure of it where ind is the index of

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

classical bounded and almost periodic solutions to quasilinear first order hyperbolic systems in a strip abstract we consider boundary value problems for quasilinear first order one dimensional hyperbolic systems in a strip the boundary conditions are supposed to be of a smoothing type in the sense that the l 2 generalized solutions to the initial boundary value problems become eventually c 2 smooth for any initial l 2 data we investigate small global classical solutions and obtain the existence and uniqueness result under the condition that the evolution family generated by the linearized problem has exponential dichotomy on r we prove that the dichotomy survives under small perturbations in the leading coefficients of the hyperbolic system assuming that the coefficients of the hyperbolic system are almost periodic we prove that the bounded solution is almost periodic also

exponential dichotomies in intermediate spaces with applications to a diffusively perturbed predator prey model abstract intermediate spaces are useful for obtaining maximal regularity when studying reaction diffusion equations we give a definition of exponential dichotomies in these spaces roughness of the exponential dichotomies and other basic properties are also proved we then use these notions to investigate the stability of a periodic orbit in a predator prey model studied by conway and smoller when diffusion terms are added

a game theoretic algorithm to compute local stabilizing solutions to hjbi equations in nonlinear h control in this paper an iterative algorithm to solve hamilton jacobi bellman isaacs hjbi equations for a broad class of nonlinear control systems is proposed by constructing two series of nonnegative functions we replace the problem of solving an hjbi equation by the problem of solving a sequence of hamilton jacobi bellman hjb equations whose solutions can be approximated recursively by existing methods the local convergence of the algorithm and local quadratic rate of convergence of the algorithm are guaranteed and a proof is given numerical examples are also provided to demonstrate the effectiveness of the proposed algorithm a game theoretical interpretation of the algorithm is given

approximating lyapunov exponents and sacker sell spectrum for retarded functional differential equations we consider lyapunov exponents and sacker sell spectrum for linear nonautonomous retarded functional differential equations posed on an appropriate hilbert space a numerical method is proposed to approximate such quantities based on the reduction to finite dimension of the evolution family associated to the system to which a classic discrete qr method is then applied the discretization of the evolution family is accomplished by a combination of collocation and generalized fourier projection a rigorous error analysis is developed to bound the difference between the computed stability spectra and the exact stability spectra the efficacy of the results is illustrated with some numerical examples

neural network based online simultaneous policy update algorithm for solving the hji equation in nonlinear h_ infty control it is well known that the nonlinear h state feedback control problem relies on the solution of the hamilton jacobi isaacs hji equation which is a nonlinear partial differential equation that has proven to be impossible to solve analytically in this paper a neural network nn based online simultaneous policy update algorithm spua is developed to solve the hji equation in which knowledge of internal system dynamics is not required first we propose an online spua which can be viewed as a reinforcement learning technique for two players to learn their optimal actions in an unknown environment the proposed online spua updates control and disturbance policies simultaneously thus only one iterative loop is needed second the convergence of the online spua is established by proving that it is mathematically equivalent to newton s method for finding a fixed point in a banach space third we develop an actor critic structure for the implementation of the online spua in which only one critic nn is needed for approximating the cost function and a least square method is given for estimating the nn weight parameters finally simulation studies are provided to demonstrate the effectiveness of the proposed algorithm

nonlinear criteria for the existence of the exponential trichotomy in infinite dimensional spaces abstract in this paper we obtain for the first time nonlinear conditions for the existence of the exponential trichotomy of skew product flows in infinite dimensional spaces we treat the most general case without any additional assumptions concerning the cocycle and without assuming a priori the existence of the projection families we show that an inedit assembly of integral conditions imply the existence of the exponential trichotomy with all of its properties and we prove that the imposed conditions are also necessary our results generalize the previous studies on this topic and provide as particular cases many interesting situations among which we mention the detection of the exponential trichotomy of general non autonomous systems

on the asymptotic behavior of discrete dynamical systems an ergodic theory approach abstract the aim of this paper is to present a new and general method for the study of stability expansiveness and dichotomy of dynamical systems based on arguments proceeding from ergodic theory and on admissibility properties we consider a discrete variational system and we associate to it three different families of input output systems corresponding to the investigations for stability expansiveness and dichotomy respectively we introduce some input output conditions relative to ergodic measures and we show that certain nonuniform admissibility properties imply the existence of the uniform asymptotic behaviors first we obtain characterizations for uniform exponential stability as an application we prove that in order to preserve the uniform exponential stability subject to perturbations it is sufficient for the perturbed system to be close enough to the initial system on a measurable set next we deduce criteria for uniform exponential expansiveness based on some exact admissibilities relative to ergodic measures finally we present an interesting application to the study of the uniform exponential dichotomy starting from a nonuniform admissibility property and using stability criteria

dichotomies for linear evolutionary equations in banach spaces abstract in this paper we present a characterization for the existence of an exponential dichotomy for a linear evolutionary system on a banach space the theory we present here applies to general time varying linear equations in banach spaces as a result it gives a description of the behavior of the nonlinear dynamics generated by certain nonlinear evolutionary equations in the vicinity of a compact invariant set in the case of dissipative systems our theory applies to the study of the flow in the vicinity of the global attractor the theory formulated here holds for linear evolutionary systems which are uniformly contracting and applies to the study of the linearization of nonlinear equations of the following type a parabolic pdes including systems of reaction diffusion equations and the navier stokes equations b hyperbolic pdes including the nonlinear wave equation and the nonlinear schrodinger equation with dissipation c retarded differential equations and d certain neutral differential delay equations

on a state space approach to nonlinear h control we study the standard h optimal control problem using state feedback for smooth nonlinear control systems the main theorem obtained roughly states that the l2 induced norm from disturbances to inputs and outputs can be made smaller than a constant 0 if the corresponding h norm for the system linearized at the equilibrium can be made smaller than by linear state feedback necessary and sufficient conditions for the latter problem are by now well known e g from the state space approach to linear h optimal control our approach to the nonlinear h optimal control problem generalizes the state space approach to the linear h problem by replacing the hamiltonian matrix and corresponding riccati equation as used in the linear context by a hamiltonian vector field together with a hamiltonian jacobi equation corresponding to its stable invariant manifold

evolution semigroups translation algebras and exponential dichotomy of cocycles abstract we study the exponential dichotomy of an exponentially bounded strongly continuous cocycle over a continuous flow on a locally compact metric space acting on a banach space x our main tool is the associated evolution semigroup on c 0 x we prove that the cocycle has exponential dichotomy if and only if the evolution semigroup is hyperbolic if and only if the imaginary axis is contained in the resolvent set of the generator of the evolution semigroup to show the latter equivalence we establish the spectral mapping annular hull theorem for the evolution semigroup in addition dichotomy is characterized in terms of the hyperbolicity of a family of weighted shift operators defined on c 0 z x here we develop banach algebra techniques and study weighted translation algebras that contain the evolution operators these results imply that dichotomy persists under small perturbations of the cocycle and of the underlying compact metric space also exponential dichotomy follows from pointwise discrete dichotomies with uniform constants finally we extend to our situation the classical perron theorem which says that dichotomy is equivalent to the existence and uniqueness of bounded continuous mild solutions to the inhomogeneous equation

roughness of tempered exponential dichotomies for infinite dimensional random difference equations abstract in this paper we study the roughness of tempered exponential dichotomies for linear random dynamical systems in banach spaces such a dichotomy has a tempered bound and describes nonuniform hyperbolicity we prove the roughness without assuming their invertibility and the integrability condition of the multiplicative ergodic theorem we give an explicit bound for the linear perturbation such that the dichotomy is persistent we also obtain explicit forms for the exponent and the bound of tempered exponential dichotomy of the perturbed random system in terms of the original ones and the perturbations

time domain fundamental solution to biot s complete equations of dynamic poroelasticity part i two dimensional solution abstract this paper develops transient fundamental solution for biot s full dynamic two dimensional equations of poroelasticity an explicit well posed laplace transform domain fundamental solution is obtained for the governing differential equations which are established in terms of solid displacements and fluid pressure in some limiting cases the solutions are shown to reduce to those of classical elastodynamics and steady state poroelasticity thus ensuring the validity of our result the closed form transient fundamental solutions both for the limiting case early time approximation and for the general case are derived from corresponding ones in the laplace transform domain by means of laplace transform techniques they represent the very first satisfactory fundamental solutions in the real time domain for full dynamic poroelasticity some characteristics of the solutions are investigated selected numerical results are presented to demonstrate the features of the waves and the accuracy of the solution is established by comparing them with laplace transform domain solutions to complete the ingredients for developing the bem the betti s reciprocal theorem is extended to the full dynamic poroelasticity based on which a time domain boundary integral equation is obtained

successive galerkin approximation algorithms for nonlinear optimal and robust control nonlinear optimal control and nonlinear h infinity control are two of the most significant paradigms in nonlinear systems theory unfortunately these problems require the solution of hamilton jacobi equations which are extremely difficult to solve in practice to make matters worse approximation techniques for these equations are inherently prone to the so called curse of dimensionality while there have been many attempts to approximate these equations solutions resulting in closed loop control with well defined stability and robustness have remained elusive this paper describes a recent breakthrough in approximating the hamilton jacobi bellman and hamilton jacobi isaacs equations successive approximation and galerkin approximation methods are combined to derive a novel algorithm that produces stabilizing closed loop control laws with well defined stability regions in addition we show how the structure of the algorithm can be exploited to reduce the amount of computation from exponential to po

existence and robustness of exponential dichotomy of linear skew product semiflows over semiflows in this paper we investigate the exponential dichotomy of linear skew product semiflows over semiflows by considering the operators generated by the integral equation related to strongly continuous cocycles over metric spaces acting on banach bundles we characterize the existence of exponential dichotomy by properties of these operators and use this characterization to prove the robustness of exponential dichotomy

evolutionary semigroups and dichotomy of linear skew product flows on locally compact spaces with banach fibers abstract we study evolutionary semigroups generated by a strongly continuous semi cocycle over a locally compact metric space acting on banach fibers this setting simultaneously covers evolutionary semigroups arising from non autonomous abstract cauchy problems and c 0 semigroups and linear skew product flows the spectral mapping theorem for these semigroups is proved the hyperbolicity of the semigroup is related to the exponential dichotomy of the corresponding linear skew product flow to this end a banach algebra of weighted composition operators is studied the results are applied in the study of roughness of the dichotomy dichotomy and solutions of nonhomogeneous equations green s function for a linear skew product flow pointwise dichotomy versus global dichotomy and evolutionary semigroups along trajectories of the flow

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

an a posteriori kam theorem for whiskered tori in hamiltonian partial differential equations with applications to some ill posed equations the goal of this paper is to develop a kam theory for tori with hyperbolic directions which applies to hamiltonian partial differential equations even to some ill posed ones the main result has an a posteriori format that is we show that if there is an approximate solution of an invariance equation which also satisfies some non degeneracy conditions then there is a true solution nearby this allows besides dealing with the quasi integrable case for the validation of numerical computations or formal perturbative expansions as well as for obtaining quasi periodic solutions in degenerate situations the a posteriori format also has other automatic consequences smooth dependence on parameters bootstrap of regularity etc we emphasize that the non degeneracy conditions required are just quantities evaluated on the approximate solution no global assumptions on the system such as twist hence they are readily verifiable in perturbation expansions we will pay attention to the quantitative relations between the sizes of the approximation and the non degeneracy conditions this will allow us to prove what experts call small twist theorems the non degeneracy conditions vanishes as the perturbation goes to zero but much slower than the error of the approximation the method of proof is based on an iterative method for solving a functional equation for the parameterization of the torus expressing that the range of the parameterization admits an evolution and is invariant we also solve functional equations for bundles which imply that are invariant under the linearization the iterative method does not use transformation theory nor action angle variables the main result does not assume that the system is close to integrable more surprisingly we do not need that the equations we study define an evolution for all initial conditions and are well posed even if the systems we study do not admit solutions for all initial conditions we show that there is a systematic way to choose initial conditions on which one can define an evolution which is quasi periodic we first develop an abstract theorem then we show how this abstract result applies to some concrete examples the examples considered in this paper are the scalar boussinesq equation and the boussinesq system both are pde models that aim to describe water waves in the long wave limit for these equations we construct small amplitude time quasi periodic solutions which are even in the spatial variable the strategy for the abstract theorem is inspired by that in fontich et al electron res announc math sci 16 9 22 2009 j differ equ 246 8 3136 3213 2009 the main part of the paper is to study infinite dimensional analogues of dichotomies which applies even to ill posed equations and which is stable under addition of unbounded perturbations this requires that we assume smoothing properties we also present very detailed bounds on the change of the splittings under perturbations

a zabczyk type method for the study of the exponential trichotomy of discrete dynamical systems the aim of this paper is to present new discrete time characterizations for uniform exponential trichotomy of dynamical systems we obtain necessary and sufficient conditions for the existence of uniform exponential trichotomy of discrete dynamical systems using a set of computational conditions of zabczyk type the main results are applied to deduce new criteria for the detection of the uniform exponential trichotomy of dynamical systems modeled by skew product flows

schaffer spaces and uniform exponential stability of linear skew product semiflows we study the exponential stability of linear skew product semiflows on locally compact metric space with banach fibers our main tool is the admissibility of a pair of the so called schaffer spaces this characterization is a very general one it includes as particular cases many interesting situations among them we can mention some results due to clark datko latushkin van minh montgomery smith randolph rabiger schnaubelt

existence of nonuniform exponential dichotomies and a fredholm alternative abstract we consider linear equations x a t x in a banach space with nonuniform exponential dichotomies in r and r and we give an admissibility condition for the existence of nonuniform exponential dichotomies in the whole line under sufficiently small perturbations we also obtain a fredholm alternative for the existence of bounded solutions under nonlinear perturbations

computing the positive stabilizing solution to algebraic riccati equations with an indefinite quadratic term via a recursive method an iterative algorithm to solve algebraic riccati equations with an indefinite quadratic term is proposed the global convergence and local quadratic rate of convergence of the algorithm are guaranteed and a proof is given numerical examples are also provided to demonstrate the superior effectiveness of the proposed algorithm when compared with methods based on finding stable invariant subspaces of hamiltonian matrices a game theoretic interpretation of the algorithm is also provided