a posteriori error control for distributed elliptic optimal control problems with control constraints discretized by hp finite elements abstract a distributed elliptic control problem with control constraints is considered which is formulated as a three field problem and consists of two variational equations for the state and the co state variables as well as of a variational inequality for the control variable two discretization approaches with h p finite elements are discussed in the first discretization approach all variables state co state and control are discretized a semi smooth newton method is introduced for solving the resulting algebraic system in the second discretization approach only the state and the co state variables are discretized whereas the control is determined by projection a simple fixed point scheme is presented for the iterative solution of this approach the main focus of the paper is on the derivation of reliable and efficient a posteriori error estimates which enables h p adaptive mesh refinements in particular the estimates can be applied to the iteration solutions so that they can be used as a stopping criterion of the iterative solution schemes in several numerical experiments the order of convergence of the adaptive discretization approaches and the efficiency as well as the reliability of the a posteriori error estimates are studied

augemented lagrangian techniques for elliptic state constrained optimal control problems we propose augmented lagrangian methods to solve state and control constrained optimal control problems the approach is based on the lagrangian formulation of nonsmooth convex optimization in hilbert spaces developed in k ito and k kunisch augmented lagrangian methods for nonsmooth convex optimization in hilbert spaces preprint 1994 we investigate a linear optimal control problem with a boundary control function as in m bergounioux numer funct anal optim 14 1993 pp 515 543 both the equation and the constraints are augmented the proposed methods are general and can be adapted to a much wider class of problems

statistical control for performance shaping using cost cumulants the cost cumulant statistical control method optimizes the system performance by shaping the probability density function of the random cost function for a stochastic system a typical optimal control method minimizes the mean first cumulant of the cost function however there are other statistical properties of the cost function such as variance second cumulant skewness third cumulant and kurtosis fourth cumulant which affect system performance in this technical note we extend the theory of traditional stochastic control by deriving the hamilton jacobi bellman hjb equation as the necessary conditions for optimality furthermore we derived the verification theorem which is the sufficient condition for higher order statistical control and construct the optimal controller in addition we utilize neural networks to numerically solve hjb partial differential equations finally we provide simulation results for an oscillator system to demonstrate our method

improved optimal conditions and iterative parameters for the optimal control problems with an integral constraint in square a distributed optimal control problem is considered with an inequality constraint on the state variable and the constraint reads that the integral of the state is not more than a given positive constant an efficient approach is introduced to investigate optimality conditions of this problem based on the uzawa algorithm an efficient algorithm is designed and its convergence is discussed with details especially the bounds of two iterative parameters are investigated with the legendre galerkin spectral method numerical results show that the algorithm is highly feasible

the equivalence between infinite horizon optimal control of stochastic systems with exponential of integral performance index and stochastic differential games a new method based on the theory of large deviations from the invariant measure is introduced for the analysis of stochastic systems with an infinite horizon exponential of integral performance index it is shown that the infinite horizon optimal exponential of integral stochastic control problem is equivalent to a stationary stochastic differential game for an auxiliary system as an application of the developed technique the infinite horizon risk sensitive lqg problem is analyzed for both the completely observed and partially observed case

an interior point algorithm with inexact step computation in function space for state constrained optimal control we consider an interior point method in function space for pde constrained optimal control problems with state constraints our emphasis is on the construction and analysis of an algorithm that integrates a newton path following method with adaptive grid refinement this is done in the framework of inexact newton methods in function space where the discretization error of each newton step is controlled by adaptive grid refinement in the innermost loop this allows to perform most of the required newton steps on coarse grids such that the overall computational time is dominated by the last few steps for this purpose we propose an a posteriori error estimator for a problem suited norm

a partition of unity method for a class of fourth order elliptic variational inequalities abstract we consider a partition of unity method pum for a class of fourth order elliptic variational inequalities on convex polygonal domains that include obstacle problems of simply supported kirchhoff plates and elliptic distributed optimal control problems with pointwise state constraints as special cases by including singular functions in the local approximation spaces we are able to show that the partition of unity method converges optimally numerical results that corroborate the theoretical estimates are also presented

error estimates for spectral approximation of elliptic control problems with integral state and control constraints abstract the goal of this paper is to investigate the legendre galerkin spectral approximation of elliptic optimal control problems with integral state and control constraints thanks to the appropriate base functions of the discrete spaces the discrete system is with sparse coefficient matrices we first present the optimality conditions of the control system then a priori and a posteriori error estimates both in h 1 and l 2 norms are derived some numerical tests indicate that the spectral accuracy can be achieved and the proposed method is competitive for solving control problems

finite element approximations of an optimal control problem with integral state constraint an integral state constrained optimal control problem governed by an elliptic partial differential equation and its finite element approximation are considered the finite element approximation is constructed on multimeshes an l 2 norm a priori error estimate of the finite element approximation is obtained further some superconvergence results are proved based on these superconvergence results almost optimal l infty norm error estimates are derived some recovery algorithms are then proposed to produce a posteriori error estimators of gradient type to solve the finite element system a simple and yet efficient iterative gradient projection algorithm is proposed and its convergence rate is proved some numerical examples are performed to confirm theoretical analysis

convergence of a finite element approximation to a state constrained elliptic control problem we consider an elliptic optimal control problem with pointwise state constraints the cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of linear finite elements and enforcing the state constraints in the nodes of the triangulation the corresponding minima are shown to converge in l 2 to the exact control as the discretization parameter tends to zero furthermore error bounds for the control and the state are obtained in both two and three space dimensions finally we present numerical examples which confirm our analytical findings

can one hear the shape of a graph we show that the spectrum of the schroperator on a finite metric graph determines uniquely the connectivity matrix and the bond lengths provided that the lengths are non commensurate and the connectivity is simple no parallel bonds between vertices and no loops connecting a vertex to itself that is one can hear the shape of the graph we also consider a related inversion problem a compact graph can be converted into a scattering system by attaching to its vertices leads to infinity we show that the scattering phase determines uniquely the compact part of the graph under similar conditions as above

risk sensitive control and dynamic games for partially observed discrete time nonlinear systems solves a finite horizon partially observed risk sensitive stochastic optimal control problem for discrete time nonlinear systems and obtains small noise and small risk limits the small noise limit is interpreted as a deterministic partially observed dynamic game and new insights into the optimal solution of such game problems are obtained both the risk sensitive stochastic control problem and the deterministic dynamic game problem are solved using information states dynamic programming and associated separated policies a certainty equivalence principle is also discussed the authors results have implications for the nonlinear robust stabilization problem the small risk limit is a standard partially observed risk neutral stochastic optimal control problem

equivalence between isospectrality and isolength spectrality for a certain class of planar billiard domains isospectrality of the planar domains which are obtained by successive unfolding of a fundamental building block is studied in relation to isolength spectrality of the corresponding domains although an explicit and exact trace formula such as poisson s summation formula or selberg s trace formula is not known to exist for such planar domains equivalence between isospectrality and isolength spectrality in a certain setting can be proved by employing the matrix representation of transplantation of eigenfunctions as an application of the equivalence transplantable pairs of domains which are all isospectral pair of planar domains and therefore counter examples of kac s question can one hear the shape of a drum are numerically enumerated and it is found that at least up to the domain composed of 13 building blocks transplantable pairs coincide with those constructed by the method due to sunada

nash equilibria in risk sensitive dynamic games dynamic games in which each player has an exponential cost criterion are referred to as risk sensitive dynamic games in this note nash equilibria are considered for such games feedback risk sensitive nash equilibrium solutions are derived for two person discrete time linear quadratic nonzero sum games both under complete state observation and shared partial observation

regularization for state constrained optimal control problems by half spaces based decoupling abstract in this paper we study an abstract constrained optimization problem which subsumes a common model for the optimal control of linear partial differential equations our emphasis is the case when the ordering cone for the optimization problem has an empty interior to circumvent this difficulty we propose a new regularization approach which is based on the decoupling of the ordering cone by using certain half spaces the approach results in a family of regular optimization problems having a simpler structure than the original optimization problem existence theorems convergence analysis and optimality conditions are given for the regularized problems we present two numerical examples and the results are quite encouraging showing the potential of the proposed approach

a noncooperative game approach to osnr optimization in optical networks we consider a game theory framework for power control in optical networks channel optical signal to noise ratio osnr optimization is formulated as an m player noncooperative game based on a general network osnr model conditions for existence and uniqueness of the game equilibrium solution are given an iterative algorithm for power control is proposed that uses only channel specific feedback measurements and is shown to converge to the equilibrium solution

on the structure of lagrange multipliers for state constrained optimal control problems abstract state constrained optimal control problems represent severe analytical and numerical challenges it is shown that under certain conditions on the active set the measure representing the lagrange multiplier associated to the state constraint can be decomposed in a distributed part in l2 with support on the active set and a boundary measure concentrated on the interface between active and inactive sets

can one hear the shape of a drum revisited a famous inverse problem posed by m kac can one hear the shape of a drum is concerned with isospectrality of drums or planer billiards and the first counter example was constructed by gordon webb and wolpert 1992 invent math 110 1 here we present pieces of numerical evidence showing that one can distinguish isospectral drums by measuring the scattering poles of exterior neumann problems this is based on the observation that the fredholm determinant appearing in the boundary element method admits a factorization into interior and exterior parts

control of an elliptic problem with pointwise state constraints this paper deals with a quadratic control problem for elliptic equations with pointwise state constraints existence and uniqueness of the solution is proved optimality conditions are given and regularity of the optimal solution is investigated

optimal control of semilinear multistate systems with state constraints this paper deals with state constrained optimal control problems governed by a semilinear multistate equation the authors prove the existence of solutions and derive optimality conditions

brief paper a neural network solution for fixed final time optimal control of nonlinear systems in this paper fixed final time optimal control laws using neural networks and hjb equations for general affine in the input nonlinear systems are proposed the method utilizes kronecker matrix methods along with neural network approximation over a compact set to solve a time varying hjb equation the result is a neural network feedback controller that has time varying coefficients found by a priori offline tuning convergence results are shown the results of this paper are demonstrated on an example

a superconvergent hybridizable discontinuous galerkin method for dirichlet boundary control of elliptic pdes we begin an investigation of hybridizable discontinuous galerkin hdg methods for approximating the solution of dirichlet boundary control problems governed by elliptic pdes these problems can involve atypical variational formulations and often have solutions with low regularity on polyhedral domains these issues can provide challenges for numerical methods and the associated numerical analysis we propose an hdg method for a dirichlet boundary control problem for the poisson equation and obtain optimal a priori error estimates for the control specifically under certain assumptions for a 2d convex polygonal domain we show the control converges at a superlinear rate we present 2d and 3d numerical experiments to demonstrate our theoretical results

on the boundary element method for billiards with corners the boundary element method is one of the reliable numerical schemes to solve the eigenvalue problem of the helmholtz equation which is justified by the fredholm theory for domains with a smooth boundary when a domain has corners however the corresponding integral equation is singular so that the boundary element method lacks its well established base employing a cutoff technique we here formulate a well grounded version of the boundary element method and also give a certain justification to the standard boundary element method even for domains with corners

statistical control of control affine nonlinear systems with nonquadratic cost functions hjb and verification theorems in statistical control the cost function is viewed as a random variable and one optimizes the distribution of the cost function through the cost cumulants we consider a statistical control problem for a control affine nonlinear system with a nonquadratic cost function using the dynkin formula the hamilton jacobi bellman equation for the nth cost moment case is derived as a necessary condition for optimality and corresponding sufficient conditions are also derived utilizing the nth moment results the higher order cost cumulant hamilton jacobi bellman equations are derived in particular we derive hjb equations for the second third and fourth cost cumulants even though moments and cumulants are similar mathematically in control engineering higher order cumulant control shows a greater promise in contrast to cost moment control we present the solution for a control affine nonlinear system using the derived hamilton jacobi bellman equation which we solve numerically using a neural network method

spectral duality for planar billiards for a bounded open domain with connected complement inr2 and piecewise smooth boundary we consider the dirichlet laplacian on and the s matrix on the complement c we show that the on shell s matricessk have eigenvalues converging to 1 ask k0 exactly when has an eigenvalue at energyk02 this includes multiplicities and proves a weak form of transparency atk k0 we also show that stronger forms of transparency such assk0 having an eigenvalue 1 are not expected to hold in general

optimization of spectral functions of dirichlet laplacian eigenvalues we consider the shape optimization of spectral functions of dirichlet laplacian eigenvalues over the set of star shaped symmetric bounded planar regions with smooth boundary the regions are represented using fourier cosine coefficients and the optimization problem is solved numerically using a quasi newton method the method is applied to maximizing two particular nonsmooth spectral functions the ratio of the nth to first eigenvalues and the ratio of the nth eigenvalue gap to first eigenvalue both of which are generalizations of the payne polya weinberger ratio the optimal values and attaining regions for n 13 are presented and interpreted as a study of the range of the dirichlet laplacian eigenvalues for both spectral functions and each n the optimal attaining region has multiplicity two nth eigenvalue

error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem abstract in this paper we study the fully discrete mixed finite element methods for quadratic convex optimal control problem governed by semilinear parabolic equations the space discretization of the state variable is done using usual mixed finite elements whereas the time discretization is based on difference methods the state and the co state are approximated by the lowest order raviart thomas mixed finite element spaces and the control is approximated by piecewise constant elements by applying some error estimates techniques of mixed finite element methods we derive a priori error estimates both for the coupled state and the control approximation finally we present a numerical example which confirms our theoretical results

galerkin spectral approximations for optimal control problems governed by the fourth order equation with an integral constraint on state abstract an integral constraint on state optimal control problem governed by the first bi harmonic equation is analyzed and its legendre galerkin spectral approximations are constructed the optimality conditions of the continuous and the discretized optimal control problems are presented by the kuhn tucker condition respectively the a priori error estimates of high accuracy are obtained furthermore an efficient projection algorithm is proposed and the convergent rate is discussed with precise details ample numerical examples are carried out to validate the theoretical results

diabolical points in the spectra of triangles accidental degeneracies between energy levels e j and e j 1 of a real hamiltonian can occur generically in a family of hamiltonians labelled by at least two parameters x y energy level surfaces in e x y space have locally a double cone diabolo connection and we refer to the degeneracies themselves as diabolical points we studied the family of systems in which a particle moves freely within hard walled triangles vibrations of triangular membranes with x and y labelling two of the angles using an efficient green function technique to compute the levels we found several diabolical points for low lying levels as well as some symmetry degeneracies the lowest diabolical point occurred for levels 5 and 6 of the triangle 130 57 30 73 18 70 the conical structure was confirmed by noting that the normal derivative u of the wavefunction at a boundary point changed sign during a small circuit of the diabolical point the form of the variation of u around a circuit and the changing pattern of nodal lines of agreed with theoretical expectations an estimate of the total number of degeneracies n d j involving levels 1 through j based on the energy scaling of cone angles and the level spacing distribution gave n d j j 2 5 and our limited data support this prediction analytical theory confirmed that for thin triangles where our computational method is slow there are no degeneracies in the energy range studied

guaranteed satisfaction of inequality state constraints in pde constrained optimization abstract optimal control problems with inequality path constraints ipcs are present in several engineering problems described by partial differential equations pde we propose an algorithm to solve pde constrained dynamic optimization problems with guaranteed satisfaction of ipcs the algorithm is based on a solution of a sequence of approximated dynamic optimization problems following the strategy of fu et al automatica 2015 for the approximation the path constraint is imposed only on a set of discrete points and is thus relaxed at the same time it is restricted by an adaptive back off and by the error bound of the pde solution the approximation is iteratively improved at iterations without violation the restriction is reduced at iterations with violation the point of maximal violation is added to the discrete set under mild assumptions we prove finite termination of the algorithm we test the algorithm on an optimal grade change of a tubular reactor

least squares solutions of the hjb equation with neural network value function approximators in this paper we present an empirical study of iterative least squares minimization of the hamilton jacobi bellman hjb residual with a neural network nn approximation of the value function although the nonlinearities in the optimal control problem and nn approximator preclude theoretical guarantees and raise concerns of numerical instabilities we present two simple methods for promoting convergence the effectiveness of which is presented in a series of experiments the first method involves the gradual increase of the horizon time scale with a corresponding gradual increase in value function complexity the second method involves the assumption of stochastic dynamics which introduces a regularizing second derivative term to the hjb equation a gradual reduction of this term provides further stabilization of the convergence we demonstrate the solution of several problems including the 4d inverted pendulum system with bounded control our approach requires no initial stabilizing policy or any restrictive assumptions on the plant or cost function only knowledge of the plant dynamics in the appendix we provide the equations for first and second order differential backpropagation

a mixed multiscale finite element method for convex optimal control problems with oscillating coefficients abstract we study numerical approximation of convex optimal control problems governed by elliptic partial differential equations with oscillating coefficients since the objective functional contains flux we approximate the problems using the mixed finite element methods we first analyze the standard mixed finite element approximation schemes then motivated by the numerical simulation of the primal variable and the flux in highly heterogeneous porous media we use a multiscale mixed finite element method to solve the state equations the multiscale finite element bases are constructed by locally solving neumann boundary value problems the analysis of the approximate control problems is carried out under the assumption that the oscillating coefficients are locally periodic which allows us to use homogenization theory to obtain the asymptotic structure of the solutions although the numerical schemes are designed for the general case

a level set approach for the solution of a state constrained optimal control problem state constrained optimal control problems for linear elliptic partial differential equations are considered the corresponding first order optimality conditions in primal dual form are analyzed and linked to a free boundary problem resulting in a novel algorithmic approach with the boundary interface between the active and inactive sets as optimization variable the new algorithm is based on the level set methodology the speed function involved in the level set equation for propagating the interface is computed by utilizing techniques from shape optimization encouraging numerical results attained by the new algorithm are reported on

error estimates of the space time spectral method for parabolic control problems abstract this work is devoted to investigate the spectral approximation of optimal control of parabolic problems the space time method is used to boost high order accuracy by applying dual petrov galerkin spectral scheme in time and spectral method in space the optimality conditions are derived and the a priori error estimates indicate the convergence of the proposed method numerical tests confirm the theoretical results and show the efficiency of the method

hp spectral element approximation for integral state constrained optimal control problems governed by harmonic equations abstract in this paper an integral state constrained optimal control problem governed by first bi harmonic equation and its h p spectral element approximation are considered we first present the optimality conditions of the continuous and discrete optimal control problems respectively then a priori error estimates of h p spectral element approximation for the optimal control problem are derived rigorously moreover a posteriori error estimates are established for the coupled state and control approximations in detail such estimators can be used to construct reliable adaptive methods for optimal control problems

generalized hamilton jacobi bellman formulation based neural network control of affine nonlinear discrete time systems in this paper we consider the use of nonlinear networks towards obtaining nearly optimal solutions to the control of nonlinear discrete time dt systems the method is based on least squares successive approximation solution of the generalized hamilton jacobi bellman ghjb equation which appears in optimization problems successive approximation using the ghjb has not been applied for nonlinear dt systems the proposed recursive method solves the ghjb equation in dt on a well defined region of attraction the definition of ghjb pre hamiltonian function hjb equation and method of updating the control function for the affine nonlinear dt systems under small perturbation assumption are proposed a neural network nn is used to approximate the ghjb solution it is shown that the result is a closed loop control based on an nn that has been tuned a priori in offline mode numerical examples show that for the linear dt system the updated control laws will converge to the optimal control and for nonlinear dt systems the updated control laws will converge to the suboptimal control

interior dirichlet eigenvalue problem exterior neumann scattering problem and boundary element method for quantum billiards an explicit expression for the fredholm determinant d e appearing in the boundary element method for two dimensional quantum billiards is derived in terms of the interior dirichlet eigenenergies and exterior neumann scattering phase shifts not only in the semiclassical regime but also in all order of h bar d e admits the factorization into the interior and exterior contributions where the former has zeros at interior dirichlet eigenenergies and the latter at resonances of the neumann scattering a new aspect of the kac close quote s inverse problem is also discussed copyright ital 1997 ital the american physical society

n player statistical nash game control m th cost cumulant optimization in this note we study a statistical nash game control problem where multiple players optimize system performance by shaping the distribution of their cost functions through cost cumulants our focus is on the nonzero sum dynamic statistical nash game which generalizes the risk sensitive nash game the necessary and sufficient conditions for the nash equilibrium solution are given for the n player m th cost cumulant optimization using the hamilton jacobi bellman hjb equations in addition we derive the optimal control strategies and provide simulation results to demonstrate the performance of our method