the singular set for the composite membrane problem in this paper we study the behavior of the singular set u nabla u 0 for solutions u to the free boundary problem delta u f chi u geq 0 g chi u 0 with f 0 f x g x 0 and c0 0 such that the density assumption u 0 cap b r z geq c 0r 2 qquad forall r r 0 holds then z is isolated the density assumption can be motivated by the following example u x 1 2 quad f 2 quad g 2 quad hbox and u 0 emptyset

singular set of solutions of semilinear elliptic equations abstract we study the behavior of the singular set u u 0 for solutions u to the semilinear elliptic equation u f x u u x where is an open set in r n and f x u u a u p b u q where a b 0 and p q 0 we show that in dimension n 2 the singular set is a discrete set and if min p q 1 then a solution u satisfies an optimal growth p q near every non isolated point of the singular set also the same results are true in dimension n 3 under an n 1 dimensional density condition

dynamic integrated system optimization and parameter estimation for discrete time optimal control of nonlinear systems an algorithm for solving nonlinear discrete time optimal control problems with model reality differences is presented the technique uses dynamic integrated system optimization and parameter estimation disope which achieves the correct optimal solution in spite of deficiencies in the mathematical model employed in the optimization procedure a version of the algorithm with a linear quadratic model based problem implemented in the c programming language is developed and applied to illustrative simulation examples an analysis of the optimality and convergence properties of the algorithm is also presented

approximation design of optimal controllers for nonlinear systems with sinusoidal disturbances abstract this paper considers an infinite time optimal damping control problem for a class of nonlinear systems with sinusoidal disturbances a successive approximation approach saa is applied to design feedforward and feedback optimal controllers by using the saa the original optimal control problem is transformed into a sequence of nonhomogeneous linear two point boundary value tpbv problems the existence and uniqueness of the optimal control law are proved the optimal control law is derived from a riccati equation matrix equations and an adjoint vector sequence which consists of accurate linear feedforward and feedback terms and a nonlinear compensation term and the nonlinear compensation term is the limit of the adjoint vector sequence by using a finite term of the adjoint vector sequence we can get an approximate optimal control law a numerical example shows that the algorithm is effective and robust with respect to sinusoidal disturbances

optimal tracking control with zero steady state error for time delay systems with sinusoidal disturbances the problem of optimal tracking control with zero steady state error for linear time delay systems with sinusoidal disturbances is considered based on the internal model principle a disturbance compensator is constructed such that the system with external sinusoidal disturbances is transformed into an augmented system without disturbances by introducing a sensitivity parameter and expanding power series around it the optimal tracking control problem can be simplified into the problem of solving an infinite sum of linear optimal control series without time delay and disturbance the obtained optimal tracking control law with zero steady state error consists of accurate linear state feedback terms and a time delay compensating term which is an infinite sum of an adjoint vector series the accurate linear terms can be obtained by solving a riccati matrix equation and a sylvester equation respectively the compensation term can be approximately obtained through a recursive algorithm a numerical simulation shows that the algorithm is effective and easily implemented and the designed tracking controller is robust with respect to the sinusoidal disturbances

successive approximation procedure of optimal tracking control for nonlinear similar composite systems abstract the optimal tracking control otc problem for a class of affine nonlinear composite systems with similar structure is considered by using a modeling technique the nonlinear similar composite system is first transformed into some quasi decoupled subsystems then the high order strongly coupled nonlinear two point boundary value tpbv problem is transformed into a sequence of linear decoupled tpbv problems through a successive approximation procedure the obtained otc law consists of an accurate linear term and a nonlinear compensation term which is the limit of the adjoint vector sequence a suboptimal tracking control law is obtained by truncating a finite iterative result of the adjoint vector sequence as its nonlinear compensation term

symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes we consider the following eigenvalue optimization problem given a bounded domain and numbers 0 a 0 find a subset d of area a for which the first dirichlet eigenvalue of the operator d is as small as possible

an unstable elliptic free boundary problem arising in solid combustion we prove a regularity result for the unstable elliptic free boundary problem u u 0 0 1 related to traveling waves in a problem arising in solid combustion the maximal solution and every local minimizer of the energy are regular that is u 0 is locally an analytic surface and u u 0 u u 0 are locally analytic functions moreover we prove a partial regularity result for solutions that are non degenerate of second order here u 0 is analytic up to a set of hausdorff dimension n 2 we discuss possible singularities ams classification 35r35 35j60 35b65

the carleman approximation approach to solve a stochastic nonlinear control problem this note investigates the optimal linear quadratic control problem in the discrete time framework for stochastic systems affected by disturbances generated by a nonlinear stochastic exosystem the application of the maximum principle to nonlinear optimal control problems does not admit in general implementable solutions therefore it is worthwhile to look for finite dimensional approximation schemes the approach followed in this note is based on the degree carleman approximation of a stochastic nonlinear system applied to the exosystem and provides a real time algorithm to design an implementable control law simulations support theoretical results and show the improvements when the approximation index is increased

guaranteed cost control of affine nonlinear systems via partition of unity method we consider the problem of guaranteed cost control gcc of affine nonlinear systems in this paper firstly the general affine nonlinear system with the origin being its equilibrium point is represented as a linear like structure with state dependent coefficient matrices secondly partition of unity method is used to approximate the coefficient matrices as a result of which the original affine nonlinear system is equivalently converted into a linear like system with modeling error a gcc law is then synthesized based on the equivalent model in the presence of modeling error under certain error condition the control law ensures that the system under control is asymptotically stable as well as that a given cost function is upper bounded a suboptimal gcc law can be obtained via solving an optimization problem in terms of linear matrix inequality lmi in stead of state dependent riccati equation sdre or hamilton jacobi equations that are usually required in solving nonlinear optimal control problems finally a numerical example is provided to illustrate the validity of the proposed method

optimal output tracking control for nonlinear systems via successive approximation approach abstract this paper deals with the optimal output tracking control ootc problem of a class of nonlinear systems whose reference input to be tracked is produced by a generalized linear exosystem to solve the nonlinear ootc problem the nonlinear two point boundary value tpbv problem derived from the necessary conditions of the ootc problem is transformed into two decoupled iterative sequences of linear differential equations the ootc law obtained consists of accurate feedforward and feedback terms and a nonlinear compensation term the former can be found by solving a sylvester equation and a riccati equation and the latter can be approximated using a successive approximation approach saa a reduced order reference input observer is constructed to make the feedforward control law physically realizable a simulation example is employed to illustrate the validity of the results

uniform regularity close to cross singularities in an unstable free boundary problem we introduce a new method for the analysis of singularities in the unstable problem delta u chi u 0 which arises in solid combustion as well as in the composite membrane problem our study is confined to points of supercharacteristic growth of the solution i e points at which the solution grows faster than the characteristic invariant scaling of the equation would suggest at such points the classical theory is doomed to fail due to incompatibility of the invariant scaling of the equation and the scaling of the solution in the case of two dimensions our result shows that in a neighborhood of the set at which the second derivatives of u are unbounded the level set u 0 consists of two c 1 curves meeting at right angles it is important that our result is not confined to the minimal solution of the equation but holds for all solutions

on the singularities of a free boundary through fourier expansion in this paper we are concerned with singular points of solutions to the unstable free boundary problem delta u chi u 0 quad hbox in b 1 the problem arises in applications such as solid combustion composite membranes climatology and fluid dynamics

optimal tracking control for large scale interconnected systems with time delays this paper considers an infinite horizon optimal tracking control problem for a class of large scale interconnected systems with state time delays by using the successive approximation approach two iteration sequences of vector differential equations are constructed meanwhile the large scale interconnected system is decomposed into finite decoupled subsystems the existence and uniqueness of the optimal solution is proved as well as the convergence of the solution sequence by finite iterations of the solution sequence a suboptimal tracking control law is obtained a reduced order reference input observer is designed to make the feedforward term of the optimal tracking control law physically realizable a numerical example shows that the presented algorithm is effective and easy to implement

multiplier methods for nonlinear optimal control error estimates are derived for an augmented lagrangian approximation to an optimal control problem convex control constraints are treated explicitly while a lagrange multiplier is introduced for the nonlinear system dynamics the nonlinear optimal control problem does not fit the classical theory for estimating the error in multiplier approximations since the natural coercivity assumption is formulated in a hilbert space where the cost is not differentiable this discrepancy between the function space setting needed for coercivity and that needed for differentiability is compensated for by regularity results associated with the necessary conditions the paper concludes with an analysis of the optimal penalty parameter corresponding to a given finite element discretization

cross shaped and degenerate singularities in an unstable elliptic free boundary problem abstract we investigate singular and degenerate behavior of solutions of the unstable free boundary problem u u 0 first we construct a solution that is not of class c 1 1 and whose free boundary consists of four arcs meeting in a cross shaped singularity this solution is completely unstable repulsive from above and below which would make it hard to get by the usual methods and even numerics is nontrivial we also show existence of a degenerate solution this answers two of the open questions in the paper r monneau g s weiss an unstable elliptic free boundary problem arising in solid combustion http arxiv org abs math ap 0507315 submitted for publication

how to initialize a second class particle we identify the ballistically and diffusively rescaled limit distribution of the second class particle position in a wide range of asymmetric and symmetric interacting particle systems with established hydrodynamic behavior respectively including zero range misanthrope and many other models the initial condition is a step profile which in some classical cases of asymmetric models gives rise to a rarefaction fan scenario we also point out a model with nonconcave nonconvex hydrodynamics where the rescaled second class particle distribution has both continuous and discrete counterparts the results follow from a substantial generalization of ferrari and kipnis arguments ann inst h poincare 31 1995 143 154 for the totally asymmetric simple exclusion process the main novelty is the introduction of a signed coupling measure as initial data which nevertheless results in a proper probability initial distribution for the site of the second class particle and makes the extension possible we also reveal in full generality a very interesting invariance property of the one site marginal distribution of the process underneath the second class particle which in particular proves the intrinsicality of our choice for the initial distribution finally we give a lower estimate on the probability of survival of a second class particle antiparticle pair

suboptimal control for nonlinear systems a successive approximation approach abstract this paper presents a successive approximation approach saa designing optimal controllers for a class of nonlinear systems with a quadratic performance index by using the saa the nonlinear optimal control problem is transformed into a sequence of nonhomogeneous linear two point boundary value tpbv problems the optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence by using the finite step iteration of the nonlinear compensation sequence we can obtain a suboptimal control law simulation examples are employed to test the validity of the saa

an optimization problem for nonlinear steklov eigenvalues with a boundary potential abstract in this paper we analyze an optimization problem for the first nonlinear steklov eigenvalue plus a boundary potential with respect to the potential function which is assumed to be uniformly bounded and with fixed l 1 norm

symmetry and rigidity for the hinged composite plate problem abstract the composite plate problem is an eigenvalue optimization problem related to the fourth order operator 2 in this paper we continue the study started in 10 focusing on symmetry and rigidity issues in the case of the hinged composite plate problem a specific situation that allows us to exploit classical techniques like the moving plane method

maximal and minimal eigenvalues and their associated nonlinear equations the spectral theory of uniformly elliptic operators a under perturbations v giving rise to operators of the form hv a v x on a bounded or unbounded region such as schrodinger operators are considered suppose that v p is constrained but v is otherwise unspecified the theory of the potentials v that maximize or minimize the eigenvalues of hv is presented the optimizing potentials are typically determined by equations of the form u w x u cu u the optimization of eigenvalues also turns out to be related to the determination of the best constants in sobolev s inequality and in its one dimensional simplification to a classical oscillator problem with instanton properties

finite element approximation to the extremal eigenvalue problem for inhomogenous materials in this paper an extremal eigenvalue problem corresponding to an inhomogeneous membrane which is composed of two different materials with different densities is investigated the convergence of the finite element discretization and the error order for the smallest eigenvalue are obtained a monotonic decreasing algorithm is presented to solve the discretized problem and numerical examples are given to demonstrate the error estimation as well as the efficiency of the method

minimization of eigenvalues for a quasilinear elliptic neumann problem with indefinite weight we investigate the minimization of the positive principal eigenvalue of the problem pu m u p 2u in u 0 on over a class of sign changing weights m with m 0 it is proved that minimizers exist and satisfy a bang bang type property in dimension one we obtain a complete description of the minimizers this problem is motivated by applications from population dynamics

optimal damping control and nonlinear elliptic systems optimal control for an elliptic equation when the control is the zero order coefficient of the differential operator is considered and an optimality system is derived under certain assumptions the problem is solved completely by giving the uniqueness and the constructive existence proof for the nonlinear optimality system an extension to the case when the state equation is a particular elliptic system is considered as well

an iterative method for suboptimal control of linear time delayed systems abstract this article presents a new approach for solving the optimal control problem ocp of linear time delay systems with a quadratic cost functional the proposed method can also be used for designing optimal control time delay systems with disturbance in this study the variational iteration method vim is employed to convert the original time delay optimal control problem tdocp into a sequence of nonhomogeneous linear two point boundary value problems tpbvps the optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence the feedback term is determined by solving riccati matrix differential equation by using the finite step iteration of a nonlinear compensation sequence we can obtain a suboptimal control law finally illustrative examples are included to demonstrate the validity and applicability of the technique

optimal sliding mode control for linear time delay systems with sinusoidal disturbances this paper develops a successive approximation approach saa of optimal sliding mode control smc for linear time delay systems with sinusoidal disturbances a sequence of two point boundary value tpbv problems with both time delay and time advance terms is derived from the optimal sliding mode design according to the saa the original tpbv problems are transformed into a sequence of linear tpbv problems without delay and advance terms the obtained smc ensures that the state trajectories reach the sliding surface in finite time and remain on it thereafter the stability of the sliding mode is proved a numerical simulation is employed to verify the effectiveness of the proposed approach

a sharp bound for the ratio of the first two eigenvalues of dirichlet laplacians and extensions the first author s work was partially supported by fondecyt chile project 0132 88 and by a summer research fellowship provided by the research council of the university of missouri columbia he would like to thank the physics department and others at the universidad de chile for their hospitality during his visit in april 1990 when much of this research was completed the second author was supported in part by fondecyt chile projects 0132 88 and 1238 90 both authors also thank fritz gesztesy for general comments and encouragement

a level set method for shape optimization in semilinear elliptic problems abstract we develop a finite element based level set method for numerically solve shape optimization problems constrained by semilinear elliptic problems by combining the shape sensitivity analysis and level set method a gradient descent algorithm is proposed to solve the model problem different from solving the nonlinear hamilton jacobi equations with finite differences in traditional level set methods we solve the linear convection equation and reinitialization equation using the characteristic galerkin finite element method the methodology can handle topology as well as shape changes in both regular and irregular design regions numerical results are presented to demonstrate the effectiveness of our algorithm as well as to verify symmetry preserving and breaking properties of optimal subdomains

delay dependent nonfragile guaranteed cost control for nonlinear time delay systems abstract this paper concerns the nonfragile guaranteed cost control problem for a class of nonlinear dynamic systems with multiple time delays and controller gain perturbations guaranteed cost control law is designed under two classes of perturbations namely additive form and multiplicative form the problem is to design a memoryless state feedback control law such that the closed loop system is asymptotically stable and the closed loop cost function value is not more than a specified upper bound for all admissible uncertainties based on the linear matrix inequality lmi approach some delay dependent conditions for the existence of such controller are derived a numerical example is given to illustrate the proposed method

optimization problems for eigenvalues of p laplace equations abstract we study minimization and maximization problems for the principal eigenvalue of a p laplace equation in a bounded domain with weight d where d is a variable subset with a fixed measure we investigate monotonicity continuity and differentiability with respect to of the optimizing eigenvalues

brief paper sequential linear quadratic control of bilinear parabolic pdes based on pod model reduction we present a framework to solve a finite time optimal control problem for parabolic partial differential equations pdes with diffusivity interior actuators which is motivated by the control of the current density profile in tokamak plasmas the proposed approach is based on reduced order modeling rom and successive optimal control computation first we either simulate the parabolic pde system or carry out experiments to generate data ensembles from which we then extract the most energetic modes to obtain a reduced order model based on the proper orthogonal decomposition pod method and galerkin projection the obtained reduced order model corresponds to a bilinear control system based on quasi linearization of the optimality conditions derived from pontryagin s maximum principle and stated as a two boundary value problem we propose an iterative scheme for suboptimal closed loop control we take advantage of linear synthesis methods in each iteration step to construct a sequence of controllers the convergence of the controller sequence is proved in appropriate functional spaces when compared with previous iterative schemes for optimal control of bilinear systems the proposed scheme avoids repeated numerical computation of the riccati equation and therefore reduces significantly the number of odes that must be solved at each iteration step a numerical simulation study shows the effectiveness of this approach

regularity of the minimizers in the composite membrane problem in r 2 we study the regularity of the minimizers to the problem a infu h01 u 2 1 d a du 2 du2 we prove that in the physical case c is analytic where c is the constant such that d u c up to a zero measure set

critical potentials of the eigenvalues and eigenvalue gaps of schrodinger operators abstract let m be a compact riemannian manifold with or without boundary and let be its laplace beltrami operator for any bounded scalar potential q we denote by i q the i th eigenvalue of the schrodinger type operator q acting on functions with dirichlet or neumann boundary conditions in case m we investigate critical potentials of the eigenvalues i and the eigenvalue gaps g i j j i considered as functionals on the set of bounded potentials having a given mean value on m we give necessary and sufficient conditions for a potential q to be critical or to be a local minimizer or a local maximizer of these functionals for instance we prove that a potential q l m is critical for the functional 2 if and only if q is smooth 2 q 3 q and there exist second eigenfunctions f 1 f k of q such that j f j 2 1 in particular 2 as well as any i admits no critical potentials under dirichlet boundary conditions moreover the functional 2 never admits locally minimizing potentials

extremal energies of laplacian operator different configurations for steady vortices abstract in this paper we study a maximization and a minimization problem associated with a poisson boundary value problem optimal solutions in a set of rearrangements of a given function define stationary and stable flows of an ideal fluid in two dimensions the main contribution of this paper is to determine the optimal solutions at first we determine a nearly optimal solution which is an approximation of the optimal solution when the problems are in low contrast regime secondly for the high contrast regime two optimization algorithms are developed for the minimization problem we prove that our algorithm converges to the global minimizer regardless of the initializer the maximization algorithm is capable of deriving all local maximizers including the global one numerical experiments lead us to a conjecture about the location of the maximizers in the set of rearrangements of a function

optimal sinusoidal disturbances damping for singularly perturbed systems with time delay abstract this paper considers an optimal damping vibration control problem for linear time delay singularly perturbed systems affected by external sinusoidal disturbances by using the slow fast decomposition theory of singularly perturbed systems the system is first decomposed into a fast subsystem without time delay and a slow subsystem with time delay affected by external sinusoidal disturbances then the successive approximation approach saa and the feedforward compensation techniques are proposed to solve the slow time scale time delay optimal control problem and damp the external sinusoidal disturbances respectively the conditions of existence and uniqueness of the finite and infinite horizon feedforward and feedback composite control ffcc laws are presented and the design approaches are given the ffcc laws consist of nondelay feedback terms disturbances compensation terms and a time delay compensation term which is the limit of the solution sequence of the adjoint vector equations simulation examples indicate that the saa is valid and the ffcc laws are easy to implement and more effective with respect to damping the external sinusoidal disturbances than that of the classical feedback composite control fcc laws