the application of generalized diagonal dominance to linear system stability theory abstract this paper shows that the well known diagonal dominance condition associated with rosenbrock s stability theorems is unnecessary and can be replaced by a less restrictive generalized diagonal dominance condition similar results are derived for generalized block diagonally dominant systems whenever possible graphical interpretations are given to facilitate the implementation of these ideas in computer aided design packages

robust stability of additively perturbed interconnected systems the concept of generalized block diagonal dominance is used to derive robust stability results for additively perturbed interconnected systems these results characterize the robustness of interconnected systems by putting upper bounds on the spectral norms of the various subblocks of admissible perturbations

existence uniqueness and regularity of solutions to systems of nonlocal obstacle problems related to optimal switching we study viscosity solutions to a system of nonlinear degenerate parabolic partial integrodi erential equations with interconnected obstacles this type of problem occurs in the context of optimal

approximation schemes for viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of applications including the calculus of variations control theory and differential games recently m g crandall and p l lions established the correct notion of generalized solutions for these equations this article discusses the convergence of general approximation schemes to this solution and gives under certain hypotheses explicit error estimates these results are then applied to obtain various representations as limits of solutions of general explicit and implicit finite difference schemes with error estimates

a priori estimates of smoothness of solutions to difference bellman equations with linear and quasi linear operators a priori estimates for finite difference approximations for the first and second order derivatives are obtained for solutions of parabolic equations described in the title

an inclusion region for the eigenvalues of partitioned matrices abstract this note presents a new result which defines disc regions centred on the eigenvalues of the diagonal blocks of a general complex matrix which contain the eigenvalues of the matrix a result of this type is useful in the application of a generalized nyquist type stability criterion to interconnected systems

difference quadrature schemes for nonlinear degenerate parabolic integro pde we derive and analyze monotone difference quadrature schemes for bellman equations of controlled levy jump diffusion processes these equations are fully nonlinear degenerate parabolic integro pdes interpreted in the sense of viscosity solutions we propose new direct discretizations of the nonlocal part of the equation that give rise to monotone schemes capable of handling singular levy measures furthermore we develop a new general theory for deriving error estimates for approximate solutions of integro pdes which thereafter is applied to the proposed difference quadrature schemes

nonlinear diffusion effects on biological population spatial patterns motivated by the observation that anomalous diffusion is a realistic feature in the dynamics of biological populations we investigate its implications in a paradigmatic model for the evolution of a single species density u x t the standard model includes growth and competition in a logistic expression and spreading is modeled through normal diffusion moreover the competition term is nonlocal which has been shown to give rise to spatial patterns we generalize the diffusion term through the nonlinear form tu x t d xxu x t with d 0 encompassing the cases where the state dependent diffusion coefficient either increases 1 or decreases 1 with the density yielding subdiffusion or superdiffusion respectively by means of numerical simulations and analytical considerations we display how that nonlinearity alters the phase diagram the type of diffusion imposes critical values of the model parameters for the onset of patterns and strongly influences their shape inducing fragmentation in the subdiffusive case the detection of the main persistent mode allows analytical prediction of the critical thresholds

a population model with nonlinear diffusion abstract a model is presented for a single species population moving in a limited one dimensional environment the birth death process is specialized by assuming a constant death modulus and a birth modulus which is an exponential in the age the diffusion mechanism is nonlinear and results in a problem for the space population density which has a degenerate parabolic form and is similarly to the porous media equation it is shown that the effect of the nonlinearity in the diffusion is to produce an approach to steady state even when the process is birth dominant the interaction of the birth death and diffusion processes is studied and is shown to yield a modified birth death mechanism which is both time and space dependent

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

duality based a posteriori error estimates for some approximation schemes for optimal investment problems abstract we consider a markov chain approximation scheme for utility maximization problems in continuous time which uses in turn a piecewise constant policy approximation euler maruyama time stepping and a gaus hermite approximation of the gausian increments the error estimates previously derived in picarelli and reisinger 2018 are asymmetric between lower and upper bounds due to the control approximation and improve on known results in the literature in the lower case only in the present paper we use duality results to obtain a posteriori upper error bounds which are empirically of the same order as the lower bounds the theoretical results are confirmed by our numerical tests

optimal convergence rate of the explicit finite difference scheme for american option valuation an optimal convergence rate o dx for an explicit finite difference scheme for a variational inequality problem is obtained under the stability condition s email protected dx 2 1 using completely pde methods as a corollary a binomial tree scheme of an american put option where s email protected dx 2 1 is convergent unconditionally with the rate o dt 1 2

a multidimensional exponential utility indifference pricing model with applications to counterparty risk this paper considers exponential utility indifference pricing for a multidimensional nontraded assets model subject to intertemporal default risk and provides a semigroup approximation for the utility indifference price the key tool is the splitting method whose convergence is proved based on the barles souganidis monotone scheme and the convergence rate is derived based on krylov s shaking the coefficients technique we apply our methodology to study the counterparty risk of derivatives in incomplete markets

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

optimal switching for ordinary differential equations we consider the problem of controlling an ordinary differential equation subject to positive switching costs and show in particular that the value functions form the viscosity solution cf 6 7 of the dynamic programming quasi variational inequalities this interpretation allows for a rigorous application of various dynamic programming techniques

min max formulas for nonlocal elliptic operators on euclidean space abstract an operator satisfies the global comparison property if anytime a function touches another from above at some point then the operator preserves the ordering at the point of contact this is characteristic of degenerate elliptic operators including nonlocal and nonlinear ones in previous work the authors considered such operators in riemannian manifolds and proved they can be represented by a min max formula in terms of levy operators in this note we revisit this theory in the context of euclidean space with the intricacies of the general riemannian setting gone the ideas behind the original proof of the min max representation become clearer moreover we prove new results regarding operators that commute with translations or which otherwise enjoy some spatial regularity

overlapping decompositions expansions contractions and stability of hybrid systems inclusion and extension principles are defined and studied for hybrid systems the controller design problem for hybrid systems is then considered within the framework of the extension principle it is shown that if the extension principle is used then any controller designed in the expanded spaces is contractible to the original spaces for implementation the stability of hybrid systems is also considered a number of definitions of stability for hybrid systems are given and it is shown that if a system which includes another system is stable then the included system is also stable furthermore it is shown that if a controller designed for the expanded system achieves stability then the contracted controller also achieves stability for the original system decentralized controller design with overlapping decompositions is also discussed within the framework of extension an example is provided to illustrate the proposed approach

decentralized estimation and control with overlapping input state and output decomposition abstract the extension principle which was introduced recently is generalized to the case of input state and output expansion estimator and controller design problems are considered within the framework of the extension principle decentralized estimator and controller design with overlapping decompositions is also discussed within the same framework it is shown that if the extension principle is used then any estimator or controller designed in the expanded spaces is contractible to the original spaces for implementation furthermore it is shown that if an estimator designed for the expanded system achieves good estimation and or asymptotic estimation then the contracted estimator also achieves good estimation and or asymptotic estimation for the original system similarly it is also shown that if a controller designed for the expanded system achieves stability and or good performance then the contracted controller achieves stability and or good performance for the original system

error bounds for monotone approximation schemes for parabolic hamilton jacobi bellman equations we obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation numerical schemes for parabolic hamilton jacobi bellman equations by introducing a new notion of consistency our results are robust and general they improve and extend earlier results by krylov barles and jakobsen we apply our general results to various schemes including crank nicholson type finite difference schemes splitting methods and the classical approximation by piecewise constant controls in the first two cases our results are new and in the last two cases the results are obtained by a new method which we develop here

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

overlapping block diagonal dominance and existence of liapunov functions the main objective of this paper is to formulate a generalization of block diagonal dominance which can be used to establish nonsingularity of matrices via overlapping diagonal blocks a number of stability results are derived in the new setting by exploiting the well known m matrix properties as well as extensions of the normalization scaling and alternative norm utilization a link between generalized block diagonal dominance and vector liapunov functions is established which can be applied in the stability analysis of interconnected dynamic systems

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

an inclusion principle for dynamic systems the purpose of this paper is to present a detailed study of the inclusion concept in dynamic systems which is a suitable mathematical framework for comparing systems with different dimensions the framework offers immediate results in reduced order modeling and the overlapping decentralized control of complex systems the presentation which is limited to linear constant systems relies on both the matrix algebra computations and the geometric elements structure to provide a balanced view of the issues involved in the concept of inclusion the framework is quite broad and has been used to consider nonlinear and time varying systems as well as systems with hereditary and stochastic effects

uniqueness results for second order bellman isaacs equations under quadratic growth assumptions and applications in this paper we prove a comparison result between semicontinuous viscosity sub and supersolutions growing at most quadratically of second order degenerate parabolic hamilton jacobi bellman and isaacs equations as an application we characterize the value function of a finite horizon stochastic control problem with unbounded controls as the unique viscosity solution of the corresponding dynamic programming equation

inclusion principle for descriptor systems the purpose of this paper is to propose an expansion contraction framework for linear constant descriptor systems within the inclusion principle for dynamic systems our primary objective is to provide an explicit characterization of the expansion process whereby a given descriptor system is expanded into the larger space where all its solutions are reproducible by the expanded descriptor system if appropriate initial conditions are selected when a control law is formulated in the expanded space the proposed characterizations provide contractibility conditions for implementation of the control law in the original system a full freedom is provided for selecting appropriate matrices in the proposed expansion contraction control scheme in particular the derived theoretical framework serves as a flexible environment for expansion contraction control design of descriptor systems under overlapping information structure constraints

on the rate of convergence of the binomial tree scheme for american options an american put option can be modelled as a variational inequality with a penalization approximation to this variational inequality the convergence rate o big delta x 2 3 big of the binomial tree scheme is obtained in this paper

a semi lagrangian approach for natural gas storage valuation and optimal operation the valuation of a gas storage facility is characterized as a stochastic control problem resulting in a hamilton jacobi bellman hjb equation in this paper we present a semi lagrangian method for solving the hjb equation for a typical gas storage valuation problem the method is able to handle a wide class of spot price models that exhibit mean reverting seasonality dynamics and price jumps we develop fully implicit and crank nicolson timestepping schemes based on a semi lagrangian approach and prove the convergence of fully implicit timestepping to the viscosity solution of a modified hjb equation posed on a bounded domain provided that a strong comparison result holds the semi lagrangian approach avoids policy type iterations required by an implicit finite difference method without requiring additional cost numerical experiments are presented for several variants of the basic scheme

differential games optimal control and directional derivatives of viscosity solutions of bellman s and isaacs equations recent work by the authors and others has demonstrated the connections between the dynamic programming approach to optimal control theory and to two person zero sum differential games problems and the new notion of viscosity solutions of hamilton jacobi pde s introduced by m g crandall and p l lions in particular it has been proved that the dynamic programming principle implies that the value function is the viscosity solution of the associated hamilton jacobi bellman and isaacs equations in the present work it is shown that viscosity super and subsolutions of these equations must satisfy some inequalities called super and subdynamic programming principle respectively this is then used to prove the equivalence between the notion of viscosity solutions and the conditions introduced by a subbotin concerning the sign of certain generalized directional derivatives

dynamic pricing and periodic ordering for a stochastic inventory system with deteriorating items abstract in this paper we consider a problem of the dynamic pricing and the periodic ordering for deteriorating items with a stochastic inventory level depending on the stock dependent demand and the selling price in the model to be established both shortages and remains are allowed and the backlogging rate is variable and dependent on the waiting time for the next replenishment combined with the dynamic pricing a stochastic dynamic optimization model which maximizes the total profit is developed based on the dynamic programming principle the stochastic control model is transformed into a hamilton jacobi bellman hjb equation we show that there exists an optimal replenishment cycle length with an optimal inventory level at start of cycle moreover the optimal pricing strategies are given in addition the finite difference scheme with the semi smooth newton method is proposed to solve the hjb equation numerically and some numerical simulations are presented to illustrate the theoretical results the sensitivity analysis of the optimal solution with respect to the major parameters is carried out and some managerial insights are proposed

a frequency domain method for decentralized stabilization based on quasi block diagonal dominance in this paper decentralized stabilization of interconnected systems based on quasi block diagonal dominance of transfer function matrices has been examined some previous results are refined and extended to the case where diagonal blocks are not necessarily square it has been shown that the refined stability result leads to a necessary and sufficient condition for robust decentralized stability for a class of additively perturbed interconnected systems some design considerations for decentralized controllers are discussed

risk sensitive control on an infinite time horizon stochastic control problems on an infinite time horizon with exponential cost criteria are considered the donsker varadhan large deviation rate is used as a criterion to be optimized the optimum rate is characterized as the value of an associated stochastic differential game with an ergodic expected average cost per unit time cost criterion if we take a small noise limit a deterministic differential game with average cost per unit time cost criterion is obtained this differential game is related to robust control of nonlinear systems

uniqueness for viscosity solutions of nonstationary hamilton jacobi bellman equations under some a priori conditions with applications this paper extends the uniqueness results for viscosity solutions of nonstationary hamilton jacobi bellman equations the conditions for uniqueness which are obtained can involve a trade off between the growth of the solution and the growth of the hamiltonian in particular the result is valid for solutions which grow quadratically in the space variable and are associated with hamiltonians which also grow quadratically this particular class arises in the robust control limit of risk sensitive stochastic control problems

consistency of generalized finite difference schemes for the stochastic hjb equation we analyze a class of numerical schemes for solving the hjb equation for stochastic control problems which enters the framework of markov chain approximations and generalizes the usual finite difference method the latter is known to be monotonic and hence valid only if the scaled covariance matrix is dominant diagonal we generalize this result by given the set of neighboring points allowed to enter the scheme showing how to compute effectively the class of covariance matrices that is consistent with this set of points we perform this computation for several cases in dimensions 2 3 and 4

existence of solutions to the hamilton jacobi bellman equation under quadratic growth conditions abstract in this paper we show the existence of solutions with quadratic growth to hamilton jacobi bellman equations we assume that the hamiltonian has the quadratic growth both in x and p we obtain the explicit lipschitz bound of solutions in the weighted sup norm also we discuss the vanishing viscosity limit of solutions and show the existence of viscosity solutions to the resulting hamilton jacobi equation

on the rate of convergence of finite difference approximations for bellmans equations with variable coefficients the estimates presented here for parabolic bellman s equations with variable coefficients extend the ones earlier obtained for constant coefficients

maximal use of central differencing for hamilton jacobi bellman pdes in finance in order to ensure convergence to the viscosity solution the standard method for discretizing hamilton jacobi bellman partial differential equations uses forward backward differencing for the drift term in this paper we devise a monotone method which uses central weighting as much as possible in order to solve the discretized algebraic equations we have to maximize a possibly discontinuous objective function at each node nevertheless convergence of the overall iteration can be guaranteed numerical experiments on two examples from the finance literature show higher rates of convergence for this approach compared to the use of forward backward differencing only

krylov and safonov estimates for degenerate quasilinear elliptic pdes we here establish an a priori holder estimate of krylov and safonov type for the viscosity solutions of a degenerate quasilinear elliptic pde of non divergence form the diffusion matrix may degenerate when the norm of the gradient of the solution is small the exhibited holder exponent and holder constant only depend on the growth of the source term and on the bounds of the spectrum of the diffusion matrix for large values of the gradient in particular the given estimate is independent of the regularity of the coefficients as in the original paper by krylov and safonov the proof relies on a probabilistic interpretation of the equation

the peierls nabarro model as a limit of a frenkel kontorova model abstract we study a generalization of the fully overdamped frenkel kontorova model in dimension n 1 this model describes the evolution of the position of each atom in a crystal and is mathematically given by an infinite system of coupled first order odes we prove that for a suitable rescaling of this model the solution converges to the solution of a peierls nabarro model which is a coupled system of two pdes typically an elliptic pde in a domain with an evolution pde on the boundary of the domain this passage from the discrete model to a continuous model is done in the framework of viscosity solutions

viscosity solutions of isaacs equations and differential games with lipschitz controls abstract it is demonstrated that the upper and lower values of a two person zero sum differential game solve the respective upper and lower isaacs equations in the viscosity sense introduced by crandall and lions trans amer math soc 277 1983 1 42 since such solutions are unique this yields a fairly simple proof that the game has value should the minimax condition hold as a further application of viscosity techniques a new and simpler proof that the upper and lower values can be approximated by the values of certain games with lipschitz controls is given

decentralized blocking zeros and the decentralized strong stabilization problem this paper is concerned with a new system theoretic concept decentralized blocking zeros and its applications in the design of decentralized controllers for linear time invariant finite dimensional systems the concept of decentralized blocking zeros is a generalization of its centralized counterpart to multichannel systems under decentralized control decentralized blocking zeros are defined as the common blocking zeros of the main diagonal transfer matrices and various complementary transfer matrices of a given plant as an application of this concept we consider the decentralized strong stabilization problem dssp where the objective is to stabilize a plant using a stable decentralized controller it is shown that a parity interlacing property should be satisfied among the real unstable poles and real unstable decentralized blocking zeros of the plant for the dssp to be solvable that parity interlacing property is also sufficient for the solution of the dssp for a large class of plants satisfying a certain connectivity condition the dssp is exploited in the solution of a special decentralized simultaneous stabilization problem called the decentralized concurrent stabilization problem dcsp various applications of the dcsp in the design of controllers for large scale systems are also discussed

contractible controller design and optimal control with state and input inclusion abstract a new inclusion concept which involves both state and input expansion is introduced it is shown that with the approach undertaken here any control law designed in the expanded spaces is always contractible to the original spaces the inclusion of the cost functions is also discussed and necessary and sufficient conditions are derived it is shown that unlike the case when the two input spaces are identical the inclusion conditions must in general depend on the control laws however certain sufficient conditions which do not involve final control laws can be found if the inclusion is taken with respect to the optimal control finally controller design with overlapping decompositions is discussed within the framework of the new inclusion concept and optimal control

value in mixed strategies for zero sum stochastic differential games without isaacs condition in the present work we consider 2 person zero sum stochastic differential games with a nonlinear pay off functional which is defined through a backward stochastic differential equation our main objective is to study for such a game the problem of the existence of a value without isaacs condition not surprising this requires a suitable concept of mixed strategies which to the authors best knowledge was not known in the context of stochastic differential games for this we consider nonanticipative strategies with a delay defined through a partition pi of the time interval 0 t the underlying stochastic controls for the both players are randomized along pi by a hazard which is independent of the governing brownian motion and knowing the information available at the left time point t j 1 of the subintervals generated by pi the controls of players 1 and 2 are conditionally independent over t j 1 t j it is shown that the associated lower and upper value functions w pi and u pi converge uniformly on compacts to a function v the so called value in mixed strategies as the mesh of pi tends to zero this function v is characterized as the unique viscosity solution of the associated hamilton jacobi bellman isaacs equation

multiscale problems and homogenization for second order hamilton jacobi equations we prove a general convergence result for singular perturbations with an arbitrary number of scales of fully nonlinear degenerate parabolic pdes as a special case we cover the iterated homogenization for such equations with oscillating initial data explicit examples among others are the two scale homogenization of quasilinear equations driven by a general hypoelliptic operator and the n scale homogenization of uniformly parabolic fully nonlinear pdes

differential games and directional derivatives of viscosity solutions of issacs equations ii recent work by the authors this journal 23 1985 pp 566 583 has demonstrated the con nections between the notion of viscosity sub and super solutions of first order dynamic programming pde and the optimality principle of dynamic programming as well as the directional derivatives of viscosity solutions of the above equations at an arbitrary point the present note contains a remark and a counterexample which complement the results cited

continuous dependence results for non linear neumann type boundary value problems abstract we obtain estimates on the continuous dependence on the coefficient for second order non linear degenerate neumann type boundary value problems our results extend previous work of cockburn et al jakobsen and karlsen and gripenberg to problems with more general boundary conditions and domains a new feature here is that we account for the dependence on the boundary conditions as one application of our continuous dependence results we derive for the first time the rate of convergence for the vanishing viscosity method for such problems we also derive new explicit continuous dependence on the coefficients results for problems involving bellman isaacs equations and certain quasilinear equation

synchronization of life cycles by collective transport and emergence of supercycles a model of physiological age accompanied by nonlinear diffusion in space is studied analytically and numerically and is shown to develop nonstationary traveling population waves a window of intermediate growth rates is found where collective supercycles are formed from individual stochastic life cycles supercycle periods can be considerably different larger or smaller than the average longevities of contributing individuals while the time averaged spatial expansion rate has a local maximum in the supercycling mode a method of adiabatic similarity solutions is used to derive dependencies of the solution parameters on source and sink inhomogeneities and obtain closed coupled dynamic equations for the age structure and leading and trailing fronts analytical results are compared with numerically computed similarity and full solutions for several types of population waves we discuss possible model applications to development of lichen thallus multiyear patterns of agricultural crop yields and autocorrelation of locust swarming

ergodicity of robust switching control and nonlinear system of quasi variational inequalities we analyze the asymptotic behavior for a system of fully nonlinear parabolic and elliptic quasi variational inequalities these equations are related to robust switching control problems introduced in 3 we prove that as time horizon goes to infinity resp discount factor goes to zero the long run average solution to the parabolic system resp the limiting discounted solution to the elliptic system is characterized by a solution of a nonlinear system of ergodic variational inequalities our results hold under a dissipativity condition and without any non degeneracy assumption on the diffusion term our approach uses mainly probabilistic arguments and in particular a dual randomized game representation for the solution to the system of variational inequalities

stochastic perron s method and elementary strategies for zero sum differential games we develop here the stochastic perron method in the framework of two player zero sum differential games we consider the formulation of the game where both players play symmetrically feedback strategies as opposed to the elliott kalton formulation prevalent in the literature the class of feedback strategies we use is carefully chosen so that the state equation admits strong solutions and the technicalities involved in the stochastic perron method carry through in a rather simple way more precisely we define the game over elementary strategies which are well motivated by intuition within this framework the stochastic perron method produces a viscosity subsolution of the upper isaacs equation dominating the upper value of the game and a viscosity supersolution of the upper isaacs equation lying below the upper value of the game using a viscosity comparison result we obtain that the upper value is the unique and continuous viscosity solution of the upper isaacs equation an identical statement hol

a canonical form for the inclusion principle of dynamic systems the inclusion principle provides a mathematical framework for comparing behavior of dynamic systems having different dimensions our main objective is to derive a canonical form for larger systems expansions that are obtained by expanding smaller systems contractions the form offers full freedom in selecting appropriate matrices for the expansion contraction process we will broaden the form to include feedback and propose an explicit characterization of contractible control laws subject to overlapping information structure constraints

uniqueness for parabolic equations without growth condition and applications to the mean curvature flow in r2 abstract in this article we prove a comparison result for viscosity solutions of a certain class of fully nonlinear possibly degenerate parabolic equations the main new feature of this result is that it holds for any possibly discontinuous solutions without imposing any restrictions on their growth at infinity the main application of this result which was also our main motivation to prove it is the uniqueness of solutions to one dimensional equations including the mean curvature equation for graphs without assuming any restriction on their behavior at infinity

an inclusion principle for hereditary systems abstract given two hereditary dynamic systems having different dimensions the conditions are provided under which a part of the motion of the larger system is reproduced by the smaller system that is the larger system includes the smaller one the conditions for inclusion are useful in applying the concept of vector liapunov functions to stability analysis of systems composed of overlapping subsystems by expanding the systems into a larger space the overlapping subsystems appear as disjoint and standard methods can be used to conclude stability of the expanded system under the inclusion conditions stability of the expansion implies stability of the original system an example is provided to show stability where the standard disjoint decompositions fail

exponentially growing solutions of parabolic isaacs equations abstract this paper contributes to the literature on unbounded viscosity solutions of fully nonlinear and possibly degenerate parabolic equations under natural assumptions it is established that the initial value problem for a degenerate parabolic isaacs equation set in 0 t r n has a unique continuous viscosity solution with at most exponential growth at infinity

continuous dependence estimates and homogenization of quasi monotone systems of fully nonlinear second order parabolic equations abstract the aim of this paper is to extend the continuous dependence estimates proved by jakobsen and karlsen 2002 3 to quasi monotone systems of fully nonlinear second order parabolic equations as a by product of these estimates we get an holder estimate for bounded solutions of systems and a rate of convergence estimate for the vanishing viscosity approximation in the second part of the paper we employ similar techniques to study the periodic homogenization of quasi monotone systems of fully nonlinear second order uniformly parabolic equations finally some examples are discussed

error bounds for monotone approximation schemes for hamilton jacobi bellman equations we obtain error bounds for monotone approximation schemes of hamilton jacobi bellman equations these bounds improve previous results of krylov and the authors the key step in the proof of these new estimates is the introduction of a switching system which allows the construction of approximate almost smooth supersolutions for the hamilton jacobi bellman equation

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations using the maximum principle for semicontinuous functions 3 4 we prove a general continuous dependence on the nonlinearities estimate for bounded holder continuous viscosity solutions of fully nonlinear degenerate elliptic equations furthermore we provide existence uniqueness and holder continuity results for bounded viscosity solutions of such equations our results are general enough to encompass hamilton jacobi bellman isaacs s equations of zero sum two player stochastic dierential games an immediate conse quence of the results obtained herein is a rate of convergence for the vanishing viscosity method for fully nonlinear degenerate elliptic equations

a geometrical approach to the study of unbounded solutions of quasilinear parabolic equations in this article we are interested in the existence and uniqueness of solutions for quasilinear parabolic equations set in the whole space n we consider in particular cases when there is no restriction on the growth or the behavior of these solutions at infinity our model equation is the mean curvature equation for graphs for which ecker and huisken have shown the existence of smooth solutions for any locally lipschitz continuous initial data we use a geometrical approach which consists in seeing the evolution of the graph of a solution as a geometric motion which is then studied by the so called level set approach after determining the right class of quasilinear parabolic pdes which can be taken into account by this approach we show how the uniqueness for the original pde is related to fattening phenomena in the level set approach existence of solutions is proved using a local l bound obtained by using in an essential way the level set approach finally we apply these results to convex initial data and prove existence and comparison results in full generality i e without restriction on their growth at infinity

global stabilization via local stabilizing actions stabilization of a linear time invariant system via stabilization of its main diagonal subsystems is the underlying problem in all diagonal dominance techniques for decentralized control in these techniques as well as all nyquist based techniques sufficient conditions are obtained under the assumption that the collection of the unstable poles of all diagonal subsystems is the same as the unstable poles of the overall system we show that this assumption is by itself enough to construct a solution to the problem at least in cases where the diagonal subsystems have disjoint poles

probabilistic error analysis for some approximation schemes to optimal control problems abstract we introduce a class of numerical schemes for optimal stochastic control problems based on a novel markov chain approximation which uses in turn a piecewise constant policy approximation euler maruyama time stepping and a gaus hermite approximation of the gausian increments we provide lower error bounds of order arbitrarily close to 1 2 in time and 1 3 in space for lipschitz viscosity solutions coupling probabilistic arguments with regularization techniques as introduced by krylov the corresponding order of the upper bounds is 1 4 in time and 1 5 in space for sufficiently regular solutions the order is 1 in both time and space for both bounds finally we propose techniques for further improving the accuracy of the individual components of the approximation

value functions and the dirichlet problem for isaacs equation in a smooth domain in this paper we investigate probabilistic solutions of the dirichlet problem for the elliptic isaacs equation in a smooth bounded domain in euclidean space

decentralized control using quasi block diagonal dominance of transfer function matrices the purpose of this paper is to present a generalization of the nyquist array method to blockwise decompositions which is based upon a new version of block diagonal dominance an additional flexibility of the proposed method is in partitioning of the system matrices into disjoint as well as overlapping submatrices which increases considerably the class of control systems which can be designed via block diagonal dominance within this framework the controllers for individual subsystems can be designed independently of each other so that their union represents a decentralized controller for the overall system