state dependent scaling design for a unified approach to robust backstepping global robust stabilization of nonlinear systems is characterized as a state dependent scaling problem by choosing scaling functions recursively one can construct robust control laws for an extended class of strict feedback systems

classical solutions for the 3d quasi geostrophic system on a bounded domain abstract we revisit a model for three dimensional inviscid quasi geostrophic flow on bounded cylindrical domains introduced by the authors in novack and vasseur 2019 we prove the local in time existence of classical solutions

long time regularity of solutions of the hele shaw problem abstract in this paper we study the long time behavior of solutions of the one phase hele shaw problem without surface tension we show that after a finite time solutions of the hele shaw problem become starshaped and lipschitz continuous in space based on this observation we then prove that the free boundary become smooth in space and time with nondegenerate free boundary speed

a framework for robustness analysis of constrained finite receding horizon control a framework for robustness analysis of input constrained finite receding horizon control is presented under the assumption of quadratic upper bounds on the finite horizon costs we derive sufficient conditions for robust stability of the standard discrete time linear quadratic receding horizon control formulation this is achieved by recasting conditions for nominal and robust stability as an implication between quadratic forms lending itself to s procedure tools which are used to convert robustness questions to tractable convex conditions robustness with respect to plant model mismatch as well as for state measurement error is shown to reduce to the feasibility of linear matrix inequalities simple examples demonstrate the approach

the hele shaw asymptotics for mechanical models of tumor growth models of tumor growth now commonly used present several levels of complexity both in terms of the biomedical ingredients and the mathematical description our first goal here is to formulate a free boundary model of hele shaw type a variant including growth terms starting from the description at the cell level and passing to the stiff limit in the pressure law of state in contrast with the classical hele shaw problem here the geometric motion governed by the pressure is not sufficient to completely describe the dynamics a complete description requires the equation on the cell number density we then go on to consider a more complex model including the supply of nutrients through vasculature and we study the stiff limit for the involved coupled system

global weak solutions to the inviscid 3d quasi geostrophic equation in this article the authors prove the existence of global weak solutions to the inviscid three dimensional quasi geostrophic equation this equation models the evolution of the temperature on the surface of the earth it is widely used in geophysics and meteorology

global well posedness for the critical 2d dissipative quasi geostrophic equation we give an elementary proof of the global well posedness for the critical 2d dissipative quasi geostrophic equation the argument is based on a non local maximum principle involving appropriate moduli of continuity

robust stability analysis of time varying systems using time varying quadratic forms abstract we consider the robust stability of time varying linear systems subject to structured time varying uncertainty we provide an alternate proof of the result that robust stability holds if and only if a scaled small gain condition holds asymptotically the new proof employs an extension of the so called s procedure losslessness theorem to time varying quadratic forms which is of independent interest

a multiloop generalization of the circle criterion for stability margin analysis in order to provide a theoretical tool well suited for use in characterizing the stability margins e g gain and phase margins of multiloop feedback systems multiloop input output stability results generalizing the circle stability criterion are considered generalized conic sectors with centers and radii determined by linear dynamical operators are employed to enable an engineer to specify the stability margins which he desires as a frequency dependent convex set of modeling errors including nonlinearities gain variations and phase variations which the system must be able to tolerate in each feedback loop without instability the resulting stability criterion gives sufficient conditions for closed loop stability in the presence of such frequency dependent modeling errors even when the modeling errors occur simultaneously in all loops so for example stability is assured as loop gains and phases vary throughout a set of nonzero measure whose boundaries are frequency dependent the stability conditions yield an easily interpreted scalar measure of the amount by which a muitiloop system exceeds or falls short of its stability margin specifications

necessary and sufficient conditions of stability a multiloop generalization of the circle criterion a linear time invariant system with a vector output and a vector input is described this system is closed by an uncertain nonlinear time varying feedback the only information on this feedback is given by several integral quadratic inequalities i e the uncertainties under consideration generalize the so called conic nonlinearities necessary and sufficient frequency domain conditions of stability are obtained an advanced version of the s procedure losslessness theorem and some other tools of the absolute stability theory are used

duality bounds in robustness analysis abstract multipliers are used in stability theory to reduce conservatism and exploit structural information about system components for an important class of stability problems which gives a resulting infinite dimensional multiplier optimization a corresponding dual problem is stated the dual gives valuable information about the original problem in particular error bounds for its finite dimensional approximations 1997 elsevier science ltd

generalized mixed μ bounds for real and complex multiple block uncertainty with internal matrix structure new absolute stability results that unify and extend existing structured singular value bounds for mixed uncertainty are developed using frequency domain arguments these bounds generalize prior upper bounds for mixed by permitting the treatment of non diagonal real uncertain blocks as well as accounting for internal matrix structure in the uncertainty several specializations to well known absolute stability criteria from the classical literature are given which allow for a systematic comparison of these results in addressing the problem of robust stability for real parameter uncertainty

robust performance unknown disturbances and known fixed inputs this paper addresses the robust performance problem when the system under consideration is subjected to norm bounded time varying uncertainty different performance objectives are considered and computable bounds are obtained for the worst case performance in various norms in some important cases it is shown that these bounds are tight giving simple exact formulas for computing robust performance measures one of the important features of these formulas is that they enable the direct and exact calculation of a number of worst cast performance measures while avoiding entirely the iterations needed for computing these measures by existing methods that rely on repeated scalings and spectral radius bound evaluations

singular limit of the porous medium equation with a drift abstract we study the stiff pressure limit of a nonlinear drift diffusion equation where the density is constrained to stay below the maximal value one the challenge lies in the presence of a drift and the consequent lack of monotonicity in time in the limit a hele shaw type free boundary problem emerges which describes the evolution of the congested zone where density equals one we discuss pointwise convergence of the densities as well as the bv regularity of the limiting free boundary

existence and regularity of weak solutions to the quasi geostrophic equations in the spaces lp or dot h 1 2 in this paper we study the 2d quasi geostrophic equation with and without dissipation we give global existence results of weak solutions for an initial data in the space lp or dot h 1 2 in the dissipative case when the initial data is in lp p 2 we give a regularity result of these solutions

robustness analysis and synthesis for nonlinear uncertain systems a state space characterization of robustness analysis and synthesis for nonlinear uncertain systems is proposed the robustness of a class of nonlinear systems subject to l sub 2 bounded structured uncertainties is characterized in terms of a nonlinear matrix inequality nlmi which yields a convex feasibility problem as in the linear case scalings are used to find a lyapunov or storage function that give sufficient conditions for robust stability and performances sufficient conditions for the solvability of robustness synthesis problems are represented in terms of nlmis as well with the proposed nlmi characterizations it is shown that the computation needed for robustness analysis and synthesis is not more difficult than that for checking lyapunov stability the numerical solutions for robustness problems are approximated by the use of finite element methods or finite difference schemes and the computations are reduced to solving linear matrix inequalities unfortunately while the development in this paper parallels the corresponding linear theory the resulting computational consequences are of course not as favourable

local and global strong solutions for sqg in bounded domains abstract we prove local well posedness for the inviscid surface quasigeostrophic sqg equation in bounded domains of r 2 when fractional dirichlet laplacian dissipation is added global existence of strong solutions is obtained for small data for critical and supercritical cases global existence of strong solutions with arbitrary data is obtained in the subcritical cases

robust synthesis in sub 1 a globally optimal solution in this paper a method to synthesize controllers that achieve globally optimal robust performance within any prespecified tolerance against structured norm bounded time varying and or nonlinear uncertainty is developed the performance measure considered is the infinity to infinity induced norm of a system s transfer function the method developed utilizes linear relaxation techniques to solve the infinite dimensional nonconvex problem via finite dimensional linear programming problems

analysis of robust h 2 performance using multiplier theory in this paper the problem of determining the worst case h2 performance of a control system subject to linear time invariant uncertainties is considered a set of upper bounds on the performance is derived based on the theory of stability multipliers and the solution of an original optimal control problem the numerical issues raised by the resulting computational problems are discussed in particular newly developed interior point convex optimization methods combined with linear matrix inequalities apply very well to the fast and accurate solution of these problems the new results compare favorably with prior ones the method can be extended to other types of perturbations

global in time classical solutions to the 3d quasi geostrophic system for large initial data in this paper the authors show the existence of global in time classical solutions to the 3d quasi geostrophic system with ekman pumping for any smooth initial value possibly large this system couples an inviscid transport equation in mathbb r 3 with an equation on the boundary satisfied by the trace the proof combines the de giorgi regularization effect on the boundary z 0 similar to the so called surface quasi geostrophic equation with beale kato majda techniques to propagate regularity for z 0 a bootstrapping argument combining potential theory and littlewood paley techniques is used to strengthen the regularization effect on the trace up to the besov space b 1

homogenization of a hele shaw problem in periodic and random media we investigate the homogenization limit of a free boundary problem with space dependent free boundary velocities the problem under consideration has a well known obstacle problem transformation formally obtained by integrating with respect to the time variable by making rigorous the link between these two problems we are able to derive an explicit formula for the homogenized free boundary velocity and we establish the uniform convergence of the free boundaries

noncommuting multidimensional realization theory minimality reachability and observability a generalization of the notions of minimality reachability and observability for parameter dependent and multidimensional systems modeled by linear fractional transformations over noncommuting operators is presented the multidimensional system results developed herein are formally similar to and directly specialize to the standard results when one dimensional systems are considered

model reduction of multidimensional and uncertain systems model reduction methods are presented for systems represented by a linear fractional transformation on a repeated scalar uncertainty structure these methods involve a complete generalization of balanced realizations balanced gramians and balanced truncation model reduction with guaranteed error bounds based on solutions to a pair of linear matrix inequalities which generalize lyapunov equations the resulting reduction methods immediately apply to uncertainty simplification and state order reduction in the case of uncertain systems but also may be interpreted as state order reduction for multidimensional systems

global weak solutions for sqg in bounded domains we prove existence of global weak l2 solutions of the inviscid sqg equation in bounded domains 2017 wiley periodicals inc

on l robust stability and performance with l 2 and l perturbations ubscrpt perturbations in this paper we will consider systems with linear time invariant perturbations we will analyze robust performance in the l2 and l settings the l2 setting gives rise to the familiar case of structured singular values and a stability criterion is given by the small theorem we show that although the necessary and sufficient criterion of robust stability for the l case l stability with structured l gain bounded perturbations is the same small criterion a system with l2 gain bounded perturbations is never l stable

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

an optimization algorithm for checking feasibility of robust h infinity control problem for linear time varying uncertain systems an algorithm for checking feasibility of the robust h control problem for systems with time varying norm bounded uncertainty is suggested this algorithm is an iterative procedure on each step of

hele shaw limit for a system of two reaction cross diffusion equations for living tissues multiphase mechanical models are now commonly used to describe living tissues including tumour growth the specific model we study here consists of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous medium equation including cross reaction terms we study the incompressible limit when the pressure becomes stiff which generates a free boundary problem we establish the complementarity relation and also a segregation result several major mathematical difficulties arise in the two species case firstly the system structure makes comparison principles fail secondly segregation and internal layers limit the regularity available on some quantities to bv thirdly the aronson benilan estimates cannot be established in our context we are lead as it is classical to add correction terms this procedure requires technical manipulations based on bv estimates only valid in one space dimension another novelty is to establish an l1 version in place of the standard upper bound

robust stability with time varying structured uncertainty considers the problem of assessing robust stability in the presence of block diagonally structured time varying dynamic uncertainty it is shown that robust stability holds only if there exist constant scalings which lead to a small gain condition the notion of stability here is finite gain stability over finite energy signals in sharp contrast to the case of time invariant dynamic uncertainty this result is not limited by the number uncertainty blocks these results parallel previous results regarding finite gain stability over persistent bounded signals

optimal regularity for the no sign obstacle problem in this paper we prove the optimal regularity for a general obstacle type problem under the assumption that is where n is the newtonian potential this is the weakest assumption for which one can hope to get regularity as a by product of the regularity we are able to prove that under a standard thickness assumption on the zero set close to a free boundary point the free boundary is locally a graph close to provided f is dini this completely settles the question of the optimal regularity of this problem which has been the focus of much attention during the last two decades 2012 wiley periodicals inc

the inviscid three dimensional quasi geostrophic system on bounded domains we present a formal derivation of the inviscid three dimensional quasi geostrophic system qg from primitive equations on a bounded cylindrical domain a key point in the derivation is the treatment of the lateral boundary and the resulting boundary conditions it imposes on solutions to our knowledge these boundary conditions are new and differentiate our model from closely related models which have been the object of recent study these boundary conditions are natural for a variational problem in a particular hilbert space we construct solutions and prove an elliptic regularity theorem corresponding to the variational problem allowing us to show the existence of global weak solutions to qg

on convexified robust control synthesis in this paper we study a convexified robust control problem and its relation to gain scheduling control it reveals that the robust control synthesis condition becomes convex under a special plant structure moreover for this class of robust control problems the gain scheduling control approach from scaled small gain theorem will not provide any performance improvement over convexified robust controllers based on this observation a convexified robust control synthesis framework with stringent performance and computational efficacy is proposed

computational modeling of solid tumor growth the avascular stage in this paper we present a mathematical model for avascular tumor growth and its numerical study in two and three dimensions for this purpose we use a multiscale model using pdes to describe the evolution of the tumor cell densities in our model cell cycle regulation depends mainly on microenvironment the cancer growth of volume induces cell motion and tumor expansion according to biology cells grow against a basal membrane which interacts mechanically with the tumor we use a level set method to describe this membrane and we compute its influence on cell movement thanks to a stokes equation the evolution of oxygen diffusing from blood vessels to cancer cells and used to estimate hypoxia is given by a stationary diffusion equation solved with a penalty method the model has been applied to investigate the therapeutic benefit of anti invasive agents and constitutes now the basis of a numerical platform for tumor growth simulations

an input output approach to structured stochastic uncertainty we consider linear time invariant systems with exogenous stochastic disturbances and in feedback with structured stochastic uncertainties this setting encompasses linear systems with both additive and multiplicative noise our concern is to characterize second order properties such as mean square stability mss and performance a purely input output treatment of these systems is given without recourse to state space models and thus the results are applicable to certain classes of distributed systems we derive necessary and sufficient conditions for mss in terms of the spectral radius of a linear matrix operator whose dimension is that of the number of uncertainties rather than the dimension of any underlying state space models our condition is applicable to the case of correlated uncertainties and reproduces earlier results for uncorrelated uncertainties for cases where state space realizations are given linear matrix inequality equivalents of the input output conditions are given

structured dissipativity and absolute stability of nonlinear systems the paper considers a problem of absolute stability for a nonlinear uncertain system with structured uncertainty an s procedure for nonlinear systems is established and using this the paper shows that the absolute stability problem is equivalent to an absolute stability problem for a nonlinear uncertain system with unstructured uncertainty the paper also introduces a new notion of structured dissipativeness for nonlinear systems this notion extends the standard notion of dissipativeness in that it allows for multiple supply functions the paper shows that a nonlinear uncertain system containing structured uncertainties will be absolutely stable if and only if a corresponding structured dissipativeness condition is satisfied it is also shown that structured dissipativeness is equivalent to the standard notion of dissipativeness with a parameter dependent supply function

when is d g scaling both necessary and sufficient it is shown that the well known d g scaling upper bound of the structured singular value is a nonconservative test for robust stability with respect to certain linear time varying uncertainties

a variational inequality approach to hele shaw flow with a moving boundary this paper is concerned with the study of a mathematical model of the injection of fluid into a finite hele shaw cell the mathematical problem is one of solving laplace s equation in an unknown region whose boundary changes with time by a transformation of the dependent variable an elliptic variational inequality formulation of the moving boundary problem is obtained the variational inequality is shown to have a unique solution up to the time at which the cell is filled regularity results for the solution of the inequality are obtained by studying a penalty approximation of the inequality

exact calculation of the multiloop stability margin a mapping theorem by l a zadeh and c a desoer 1963 serves as the basis for an algorithm that computes the stability margin k sub m of diagonally perturbed multivariable feedback systems without conservatism the stability margin determination not only verifies system stability but also quantifies how much further the plan uncertainties can be extended before instability occurs the essence of the computational approach is the realization that the true image of a given domain can be approximated with arbitrary accuracy by first subdividing the domain into the requisite number of subdomains and then forming the union of the convex hulls of their images the technique is illustrated in an example with a plant model having three uncertainties in general the method is applicable to either siso or mimo lti systems whose uncertainties can be modeled with noninteracting coefficients in the open loop transfer function representations

computational complexity reduction in scaled h synthesis abstract the h control problem with constant scaling is considered this problem is known to reduce to a search for matrices satisfying linear matrix inequalities lmis and an additional nonconvex constraint recently globally convergent algorithms to solve this nonconvex feasibility problem have been proposed a major factor that determines the computational complexity is the number of nonconvex constraints this paper gives verifiable conditions on the plant under which the reduction of nonconvex constraints is possible the conditions appear to be a strong requirement but in fact they are nautrally satisfied in certain problems 1997 elsevier science ltd

a necessary and sufficient minimality condition for uncertain systems a necessary and sufficient condition is given for the exact reduction of systems modeled by linear fractional transformations lfts on structured operator sets this condition is based on the existence of a rank deficient solution to either of a pair of linear matrix inequalities which generalize lyapunov equations the notion of gramians is thus also generalized to uncertain systems as well as kalman like decomposition structures a related minimality condition the converse of the reducibility condition may then be inferred from these results and the equivalence class of all minimal lft realizations defined these results comprise the first stage of a complete generalization of realization theory concepts to uncertain systems subsequent results such as the definition of and rank tests on structured controllability and observability matrices are also given the minimality results described are applicable to multidimensional system realizations as well as to uncertain systems connections to formal powers series representations also exist

a hele shaw problem for tumor growth abstract we consider weak solutions to a problem modeling tumor growth under certain conditions on the initial data solutions can be obtained by passing to the stiff incompressible limit in a porous medium type problem with a lotka volterra source term describing the evolution of the number density of cancerous cells we prove that such limit solutions solve a free boundary problem of hele shaw type we also obtain regularity properties both for the solution and for its free boundary the main new difficulty arises from the competition between the growth due to the source which keeps the initial singularities and the free boundary which invades the domain with a regularizing effect new islands can be generated at singular times

global optimization for h sub spl infin control with constant diagonal scaling this paper considers the h sub spl infin control problem with constant diagonal scaling related to the robust control synthesis for systems with structured time varying uncertainties the problem is not convex in general and hence it is difficult to find a global solution the purpose of this paper is to provide an algorithm to find a suboptimal solution with any specified small tolerance from the globally optimal solution for the optimization problem the algorithm based on the triangle covering method is proposed the computational complexity analysis shows that its worst case order is polynomial in the inverse of the tolerance and the size of an a priori given interval of scaling with a fixed number of uncertainty blocks