spline collocation method for integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions to initial or boundary value problems of fredholm integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of a piecewise polynomial collocation method for solving such problems numerically presented numerical examples display that theoretical results are in good accordance with actual convergence rates of proposed algorithms

optimal control of higher order differential inclusions of bolza type with varying time interval abstract in this paper under the t 1 transversality condition for the problem considered a sufficient condition is formulated in particular the time optimal control problem is studied locally adjoint mappings are used for both convex and nonconvex cases furthermore the application of these results is demonstrated by solving an example

convergence of neural networks for programming problems via a nonsmooth lojasiewicz inequality this paper considers a class of neural networks nns for solving linear programming lp problems convex quadratic programming qp problems and nonconvex qp problems where an indefinite quadratic objective function is subject to a set of affine constraints the nns are characterized by constraint neurons modeled by ideal diodes with vertical segments in their characteristic which enable to implement an exact penalty method a new method is exploited to address convergence of trajectories which is based on a nonsmooth lstrokojasiewicz inequality for the generalized gradient vector field describing the nn dynamics the method permits to prove that each forward trajectory of the nn has finite length and as a consequence it converges toward a singleton furthermore by means of a quantitative evaluation of the lstrokojasiewicz exponent at the equilibrium points the following results on convergence rate of trajectories are established 1 for nonconvex qp problems each trajectory is either exponentially convergent or convergent in finite time toward a singleton belonging to the set of constrained critical points 2 for convex qp problems the same result as in 1 holds moreover the singleton belongs to the set of global minimizers and 3 for lp problems each trajectory converges in finite time to a singleton belonging to the set of global minimizers these results which improve previous results obtained via the lyapunov approach are true independently of the nature of the set of equilibrium points and in particular they hold even when the nn possesses infinitely many nonisolated equilibrium points

a novel neural network for nonlinear convex programming in this paper we present a neural network for solving the nonlinear convex programming problem in real time by means of the projection method the main idea is to convert the convex programming problem into a variational inequality problem then a dynamical system and a convex energy function are constructed for resulting variational inequality problem it is shown that the proposed neural network is stable in the sense of lyapunov and can converge to an exact optimal solution of the original problem compared with the existing neural networks for solving the nonlinear convex programming problem the proposed neural network has no lipschitz condition no adjustable parameter and its structure is simple the validity and transient behavior of the proposed neural network are demonstrated by some simulation results

on solving constrained optimization problems with neural networks a penalty method approach deals with the use of neural networks to solve linear and nonlinear programming problems the dynamics of these networks are analyzed in particular the dynamics of the canonical nonlinear programming circuit are analyzed the circuit is shown to be a gradient system that seeks to minimize an unconstrained energy function that can be viewed as a penalty method approximation of the original problem next the implementations that correspond to the dynamical canonical nonlinear programming circuit are examined it is shown that the energy function that the system seeks to minimize is different than that of the canonical circuit due to the saturation limits of op amps in the circuit it is also noted that this difference can cause the circuit to converge to a different state than the dynamical canonical circuit to remedy this problem a new circuit implementation is proposed

coisotropic deformations of associative algebras and dispersionless integrable hierarchies the paper is an inquiry of the algebraic foundations of the theory of dispersionless integrable hierarchies like the dispersionless kp and modified kp hierarchies and the universal whitham hierarchy of genus zero it stands out for the idea of interpreting these hierarchies as equations of coisotropic deformations for the structure constants of certain associative algebras it discusses the link between the structure constants and hirota s tau function and shows that the dispersionless hirota bilinear equations are within this approach a way of writing the associativity conditions for the structure constants in terms of the tau function it also suggests a simple interpretation of the algebro geometric construction of the universal whitham equations of genus zero due to krichever

classical r matrix theory for bi hamiltonian field systems this is a survey of the application of the classical r matrix formalism to the construction of infinite dimensional integrable hamiltonian field systems the main point is the study of bi hamiltonian structures appropriate constructions on poisson noncommutative and loop algebras as well as the central extension procedure are presented the theory is developed for 1 1 and 2 1 dimensional field and lattice soliton systems as well as hydrodynamic systems the formalism presented contains sufficiently many proofs and important details to make it self contained and complete the general theory is applied to several infinite dimensional lie algebras in order to construct both dispersionless and dispersive soliton integrable field systems

a predictive control scheme for systems with variable time delay this article deals with the robust control design problem for linear time invariant systems affected by variable feedback delay constraints on the system state are considered in the system description the starting point is the construction of a predictive control law which guarantees the existence of a nonempty robust positive invariant rpi set with respect to the closed loop dynamics subsequently an iterative algorithm is proposed in order to obtain an approximation of the maximal rpi set the problem can be treated in the framework of piecewise affine systems due to the explicit formulations of the control law obtained via multiparametric programming

sliding modes in solving convex programming problems sliding modes are used to analyze a class of dynamical systems that solve convex programming problems the analysis is carried out using concepts from the theory of differential equations with discontinuous right hand sides and lyapunov stability theory it is shown that the equilibrium points of the system coincide with the minimizers of the convex programming problem and that irrespective of the initial state of the system the state trajectory converges to the solution set of the problem the dynamic behavior of the systems is illustrated by two numerical examples

frobenius manifold for the dispersionless kadomtsev petviashvili equation we consider a frobenius structure associated with the dispersionless kadomtsev petviashvili equation this is done essentially by applying a continuous analogue of the finite dimensional theory in the space of schwartz functions on the line the potential of the frobenius manifold is found to be a logarithmic energy with quadratic external field following the construction of the principal hierarchy we construct a set of infinitely many commuting flows which extends the classical dkp hierarchy

stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations we prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically stable and unstable periodic solutions of the linear delay differential equation dot y t a t y t b t y t tau f t by a periodic boundary value approach this equation arises e g in the study of the numerical stability of collocation methods for computing periodic solutions of nonlinear delay equations we obtain convergence results for the standard collocation algorithm and for two variants in particular estimates of the difference between the collocation solution and the true solution are derived for the standard collocation scheme the convergence results are unconditional that is they do not require mesh ratio restrictions numerical results that support the theoretical findings are also given

a general methodology for designing globally convergent optimization neural networks we present a general methodology for designing optimization neural networks we prove that the neural networks constructed by using the proposed method are guaranteed to be globally convergent to solutions of problems with bounded or unbounded solution sets in contrast with the gradient methods whose convergence is not guaranteed we show that the proposed method contains both the gradient methods and nongradient methods employed in existing optimization neural networks as special cases based on the theoretical results of the proposed method we study the convergence and stability of general gradient models in the case of unisolated solutions using the proposed method we derive some new neural network models for a very large class of optimization problems in which the equilibrium points correspond to exact solutions and there is no variable parameter finally some numerical examples show the effectiveness of the method

pole assignment for uncertain systems in a specified disk by state feedback this paper presents a method for assigning the poles in a specified disk by state feedback for a linear discrete or continuous time uncertain system the uncertainty being norm bounded for this the quadratic d stabilizability concept which is the counterpart of quadratic stabilizability in the context of pole placement is defined and a necessary and sufficient condition for quadratic d stabilizability derived this condition expressed as a parameter dependent discrete riccati equation enables one to design the control gain matrix by solving iteratively a discrete riccati equation

h guaranteed cost control for uncertain continuous time linear systems abstract this paper deals with the h guaranteed cost control problems for continuous time uncertain systems it consist of the determination of a stabilizing state feedback gain which imposes on all possible closed loop models an h norm upper bound 0 assuming that the uncertain domain is convex bounded and the uncertain system is quadratic stabilizable with disturbance attenuation it is shown how to determine by means of a convex programming problem the global minimum of as a particular and important case for precisely known linear systems the last problem reduces to the classical h optimal control problem the results follow from the definition of a special parameter space on which the above mentioned problems are convex

classical r matrices and compatible poisson structures for lax equations on poisson algebras given a classical r matrix on a poisson algebra we show how to construct a natural family of compatible poisson structures for the hamiltonian formulation of lax equations examples for which our formalism applies include the benny hierarchy the dispersionless toda lattice hierarchy the dispersionless kp and modified kp hierarchies the dispersionless dym hierarchy etc

degenerate frobenius manifolds and the bi hamiltonian structure of rational lax equations the bi hamiltonian structure of certain multicomponent integrable systems generalizations of the dispersionless toda hierarchy is studied for systems derived from a rational lax function one consequence of having a rational rather than a polynomial lax function is that the corresponding bi hamiltonian structures are degenerate i e the metric that defines the hamiltonian structure has a vanishing determinant frobenius manifolds provide a natural setting in which to study the bi hamiltonian structure of certain classes of hydrodynamic systems some ideas on how this structure may be extended to include degenerate bi hamiltonian structures such as those given in the first part of the paper is given

on a convex parameter space method for linear control design of uncertain systems this paper presents a new procedure for continuous and discrete time linear control systems design it consists of the definition of a convex programming problem in the parameter space that when solved provides the feedback gain one of the most important features of the procedure is that additional design constraints are easily incorporated in the original formulation yielding solutions to problems that have raised a great deal of interest within the last few years this is precisely the case of the decentralized control problem and the quadratic stabilizability problem of uncertain systems with both dynamic and input uncertain matrices in this last case necessary and sufficient conditions for the existence of a linear stabilizing gain are provided and to the authors knowledge this is one of the first numerical procedures able to handle and solve this interesting design problem for high order continuous time or discrete time linear models the theory is illustrated by examples

rota baxter systems dendriform algebras and covariant bialgebras a generalisation of the notion of a rota baxter operator is proposed this generalisation consists of two operators acting on an associative algebra and satisfying equations similar to the rota baxter equation rota baxter operators of any weights and twisted rota baxter operators are solutions of the proposed system it is shown that dendriform algebra structures of a particular kind are equivalent to rota baxter systems it is shown further that a rota baxter system induces a weak pseudotwistor panaite and van oystaeyen 2015 15 which can be held responsible for the existence of a new associative product on the underlying algebra examples of solutions of rota baxter systems are obtained from quasitriangular covariant bialgebras hereby introduced as a natural extension of infinitesimal bialgebras aguiar 2000 3

the extended bigraded toda hierarchy we generalize the toda lattice hierarchy by considering n m dependent variables we construct roots and logarithms of the lax operator which are uniquely defined operators with coefficients that are series of differential polynomials in the dependent variables and we use them to provide a lax pair definition of the extended bigraded toda hierarchy generalizing 4 using r matrix theory we give the bi hamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions finally we study the dispersionless limit and its connection with a class of frobenius manifolds on the orbit space of the extended affine weyl groups of the a series defined by dubrovin and zhang 1998 compos math 111 167

classical r matrix like approach to frobenius manifolds wdvv equations and flat metrics a general scheme for the construction of flat pencils of contravariant metrics and frobenius manifolds as well as related solutions to witten dijkgraaf verlinde verlinde associativity equations is formulated the advantage is taken from the rota baxter identity and some relation being counterpart of the modified yang baxter identity from the classical r matrix formalism the scheme for the construction of frobenius manifolds is illustrated on the algebras of formal laurent series and meromorphic functions on riemann sphere

brief decentralized h controller design a matrix inequality approach using a homotopy method this paper considers a decentralized h control problem for multi channel linear time invariant systems our interest is focused on dynamic output feedback the control problem is reduced to a feasibility problem of a bilinear matrix inequality bmi the objective of this paper is to propose an algorithm for solving the bmi by using the idea of the homotopy method where the controller s coefficient matrices are deformed from full matrices defined by a centralized h controller to block diagonal matrices of specified dimensions which describe a decentralized h controller when a feasible decentralized h control problem is considered it can be expected that there always exists a centralized h controller for which the algorithm converges and presents a desired solution to find such a suitable centralized h controller random search in a parametrized set of h controllers with a proper dimension is suggested the efficiency of the proposed algorithm is demonstrated by an example

solving systems of linear equations via gradient systems with discontinuous righthand sides application to ls svm a gradient system with discontinuous righthand side that solves an underdetermined system of linear equations in the l sub 1 norm is presented an upper bound estimate for finite time convergence to a solution set of the system of linear equations is shown by means of the persidskii form of the gradient system and the corresponding nonsmooth diagonal type lyapunov function this class of systems can be interpreted as a recurrent neural network and an application devoted to solving least squares support vector machines ls svm is used as an example

on duality in problems of optimal control described by convex differential inclusions of goursat darboux type sufficient conditions of optimality are derived for convex and non convex problems with state constraints on the basis of the apparatus of locally conjugate mappings the duality theorem is formulated and the conditions under which the direct and dual problems are connected by the duality relation are searched for

static output feedback controllers stability and convexity the main objective of this paper is to solve the following stabilizing output feedback control problem given matrices a b sub 2 c sub 2 with appropriate dimensions find if one exists a static output feedback gain l such that the closed loop matrix a b sub 2 lc sub 2 is asymptotically stable it is known that the existence of l is equivalent to the existence of a positive definite matrix belonging to a convex set such that its inverse belongs to another convex set conditions are provided for the convergence of an algorithm which decomposes the determination of the aforementioned matrix in a sequence of convex programs hence this paper provides a new sufficient but not necessary condition for the solvability of the above stabilizing output feedback control problem as a natural extension we also discuss a simple procedure for the determination of a stabilizing output feedback gain assuring good suboptimal performance with respect to a given quadratic index some examples borrowed from the literature are solved to illustrate the theoretical results

absolute stability of systems under block diagonal memoryless uncertainties abstract absolute stability of systems under block diagonal uncertainties is studied a quadratic lyapunov function is used it is found that the quadratic stability condition of such a system is equivalent to that of systems with uncertainties restricted to be rank one matrices the main technique used is exact penalization in nonsmooth analysis

discrete galerkin method for fredholm integro differential equations with weakly singular kernels approximations to a solution and its derivatives of a boundary value problem of an nth order linear fredholm integro differential equation with weakly singular or other nonsmooth kernels are determined these approximations are piecewise polynomial functions on special graded grids for their finding a discrete galerkin method and an integral equation reformulation of the boundary value problem are used optimal global convergence estimates are derived and an improvement of the convergence rate of the method for a special choice of parameters is obtained to illustrate the theoretical results a collection of numerical results of a test problem is presented

ℋ2 guaranteed cost control for uncertain discrete time linear systems the 2 guaranteed cost control problem is analysed for uncertain discrete time linear systems it consists on the determination of a constant state feedback stabilizing matrix gain and an 2 norm upper bound which holds for all feasible models two different approaches are considered the first is based on the optimality conditions provided by bellmans s dynamic programming equation the second is based on the geometric properties of the aforementioned problem it is shown that it is possible to parametrize all stabilizing gains over a convex set in this special parameter space the 2 cost is found to be a linear function of the free parameters finally some relationships between these approaches are presented and numerical examples given

a variable structure convex programming based control approach for a class of uncertain linear systems a variable structure convex programming based control for a class of linear uncertain systems with accessible state is presented in this note a convex programming problem is solved on line by reformulating the problem in terms of a piecewise smooth penalty function and relying on a suitable analog variable structure system implementing the gradient procedure in this way the controlled system reference movement optimal with respect to a pre specified scalar convex cost function and a set of suitable equality and inequality constraints is generated an inner control loop aimed at the finite time exact tracking of the reference movement is also designed as a result the controlled system trajectory starting in the feasible region there remains and the optimal movement in the feasible region is proved to be an asymptotically stable equilibrium point of the controlled system

infinite dimensional frobenius manifolds underlying the toda lattice hierarchy abstract following the approach of carlet et al 2011 9 we construct a class of infinite dimensional frobenius manifolds underlying the toda lattice hierarchy which are defined on the space of pairs of meromorphic functions with possibly higher order poles at the origin and at infinity we also show a connection between these infinite dimensional frobenius manifolds and the finite dimensional frobenius manifolds on the orbit space of extended affine weyl groups of type a defined by dubrovin and zhang

frobenius manifolds from regular classical w algebras abstract we obtain polynomial frobenius manifolds from classical w algebras associated to regular nilpotent elements in simple lie algebras using the related opposite cartan subalgebras

constraint aggregation principle in convex optimization a general constraint aggregation technique is proposed for convex optimization problems at each iteration a set of convex inequalities and linear equations is replaced by a single surrogate inequality formed as a linear combination of the original constraints after solving the simplified subproblem new aggregation coefficients are calculated and the iteration continues this general aggregation principle is incorporated into a number of specific algorithms convergence of the new methods is proved and speed of convergence analyzed next dual interpretation of the method is provided and application to decomposable problems is discussed finally a numerical illustration is given

stabilizing receding horizon control of nonlinear time varying systems a receding horizon control scheme for nonlinear time varying systems is proposed which is based on a finite horizon optimization problem with a terminal state penalty the penalty is equal to the cost that would be incurred over an infinite horizon by applying a locally stabilizing linear control law to the nonlinear system assuming only stabilizability of the linearized system around the desired equilibrium the new scheme ensures exponential stability of the equilibrium as the length of the optimization horizon goes from zero to infinity the domain of attraction moves from the basin of attraction of the linear controller toward the basin of attraction of the infinite horizon nonlinear controller stability robustness in the face of system perturbations is also established

minimum seeking properties of analog neural networks with multilinear objective functions in this paper we study the problem of minimizing a multilinear objective function over the discrete set

a convex optimization approach to fixed order controller design for disturbance rejection in siso systems the problem of rejection of persistent unknown but bounded disturbances can be solved using the well known l sup 1 design approach however in spite of its success this theory suffers from the fact that the resulting controller may have arbitrarily high order even in the state feedback case in addition system performance is optimized under the assumption of zero initial conditions in this paper we propose a new approach to the problem of synthesizing fixed order controllers to optimally reject persistent disturbances the main result of the paper shows that this approach leads to a finite dimensional convex optimization problem that can be efficiently solved

frobenius manifolds and central invariants for the drinfeld sokolov bihamiltonian structures the drinfeld sokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine lie algebra we compute the complete sets of invariants of the related bihamiltonian structures with respect to the group of miura type transformations

piecewise polynomial collocation for fredholm integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions of initial or boundary value problems of linear fredholm integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of a piecewise polynomial collocation method for solving such problems numerically the main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of convergence of numerical solutions for all values of the nonuniformity parameter of the underlying grid

topological landau ginzburg theory with a rational potential and the dispersionless kp hierarchy based on the dispersionless kp dkp theory we study a topological landau ginzburg lg theory characterized by a rational potential writing the dkp hierarchy in a general form treating all the primaries in an equal basis we find that the hierarchy naturally includes the dispersionless continuous limit of toda hierarchy and its generalizations having a finite number of primaries several flat solutions of the topological lg theory are obtained in this formulation and are identified with those discussed by dubrovin we explicitly construct gravitational descendants for all the primary fields giving a residue formulae for the 3 point functions of the fields we show that these 3 point functions satisfy the topological recursion relation the string equation is obtained as the generalized hodograph solutions of the dkp hierarchy which show that all the gravitational effects to the constitutive equations 2 point functions can be renormalized into the coupling constants in the small phase space

the iterated galerkin method for linear integro differential equations abstract the iterated galerkin method of i h sloan which yields superconvergence for the approximate solution of integral equations is generalized to get similar results for the larger class of integro differential equations two approaches are investigated to obtain superconvergence in different norms moreover some additional aspects are discussed

the collocation method in the numerical solution of boundary value problems for neutral functional differential equations part ii differential equations with deviating arguments we consider the numerical solution of boundary value problems for neutral differential equations with deviating arguments by the collocation method which includes the finite element method and the spectral element method by using the results of part i siam j numer anal 53 2015 pp 2771 2793 of this paper we obtain concrete convergence theorems for this particular type of functional differential equations

on a boundary value problem for delay differential equations of population dynamics and chebyshev approximation in this paper a boundary value problem for delay differential equations of population dynamics is considered we obtain approximate solutions by using chebyshev polynomial series and newton raphson s procedure and give the error estimation the method of the error estimation has been obtained in an existence theorem proved by a part of the authors we carry out some numerical experiments by a computer language matlab

h sub 2 and h sub spl infin robust filtering for convex bounded uncertain systems investigates robust filtering design problems in h sub 2 and h sub spl infin spaces for continuous time systems subjected to parameter uncertainty belonging to a convex bounded polyhedral domain it is shown that by a suitable change of variables both designs can be converted into convex programming problems written in terms of linear matrix inequalities the results generalize the ones available in the literature to date in several directions first all system matrices can be corrupted by parameter uncertainty and the admissible uncertainty may be structured then assuming the order of the uncertain system is known the optimal guaranteed performance h sub 2 and h sub spl infin filters are proven to be of the same order as the order of the system a numerical example illustrate the theoretical results

numerical analysis of singularly perturbed delay differential equations with layer behavior in this paper we present a numerical method to solve boundary value problems for singularly perturbed differential difference equations with negative shift in recent papers the term negative shift has been used for delay siam j appl math 54 1994 249 siam j appl math 42 1982 502 siam j appl math 45 1985 687 siam j appl math 45 1985 708 siam j appl math 54 1994 273 similar bvps are associated with expected first exit time problem of the membrane potential in models for neuron and in variational problem in control theory the stability and convergence analysis of the method is discussed also the effect of small shift on the boundary layer solution in both the cases i e boundary layer on the left side as well as the right side is discussed by considering numerical experiments several numerical examples are solved using the presented method compared the computed result with exact solution and plotted the graphs of the solution of the problems

convex analysis of output feedback control problems robust stability and performance this paper addresses the problem of optimal h sub 2 control by output feedback necessary and sufficient conditions on the existence of a linear stabilizing output feedback gain are provided in terms of the intersection of a convex set and a set defined by a nonlinear real valued function the results can be easily extended to deal with linear uncertain systems where uncertainties are supposed to belong to convex bounded domains providing an h sub 2 guaranteed cost output feedback control thanks to the properties of the above mentioned function we show that under certain conditions convex programming tools can be used for numerical purposes examples illustrate the theoretical results

integrable hierarchies and the mirror model of local cp1 abstract we study structural aspects of the ablowitz ladik al hierarchy in the light of its realization as a two component reduction of the two dimensional toda hierarchy and establish new results on its connection to the gromov witten theory of local c p 1 we first of all elaborate on the relation to the toeplitz lattice and obtain a neat description of the lax formulation of the al system we then study the dispersionless limit and rephrase it in terms of a conformal semisimple frobenius manifold with non constant unit whose properties we thoroughly analyze we build on this connection along two main strands first of all we exhibit a manifestly local bi hamiltonian structure of the ablowitz ladik system in the zero dispersion limit second we make precise the relation between this canonical frobenius structure and the one that underlies the gromov witten theory of the resolved conifold in the equivariantly calabi yau case a key role is played by dubrovin s notion of almost duality of frobenius manifolds as a consequence we obtain a derivation of genus zero mirror symmetry for local c p 1 in terms of a dual logarithmic landau ginzburg model

finite element boundary value integration of wheeler feynman electrodynamics the electromagnetic two body problem is solved as a boundary value problem associated to an action functional we show that the functional is frechet differentiable and that its conditions for criticality are the mixed type neutral differential delay equations with state dependent delay of wheeler feynman electrodynamics we construct a finite element method that finds c 1 smooth solutions when suitable past and future positions of the particles are given as boundary data the numerical trajectories satisfy a variational problem defined in a finite dimensional hermite functional space of c 1 piecewise polynomials the numerical variational problem is solved using a combination of newton s method intercalated with boundary adjustments to ensure that the velocity of the solution is continuous with the boundary data we recover the known circular orbits and compute several other novel trajectories of the wheeler feynman electrodynamics we also discuss the local convexity of the functional close to the new found trajectories and the possibility of solutions with less regularity

a novel recurrent neural network for solving nonlinear optimization problems with inequality constraints this paper presents a novel recurrent neural network for solving nonlinear optimization problems with inequality constraints under the condition that the hessian matrix of the associated lagrangian function is positive semidefinite it is shown that the proposed neural network is stable at a karush kuhn tucker point in the sense of lyapunov and its output trajectory is globally convergent to a minimum solution compared with variety of the existing projection neural networks including their extensions and modification for solving such nonlinearly constrained optimization problems it is shown that the proposed neural network can solve constrained convex optimization problems and a class of constrained nonconvex optimization problems and there is no restriction on the initial point simulation results show the effectiveness of the proposed neural network in solving nonlinearly constrained optimization problems

the τ function of the universal whitham hierarchy matrix models and topological field theories the universal witham hierarchy is considered from the point of view of topological field theories the function for this hierarchy is defined it is proved that the algebraic orbits of whitham hierarchy can be identified with various topological matter models coupled with topological gravity

robust finite time stabilisation of uncertain linear systems this article deals with the problem of finite time stability and stabilisation of uncertain linear systems for linear time varying systems subject to norm bounded uncertainties some conditions for finite time stability are provided these conditions are expressed in terms of differential linear matrix inequalities then the problem of controller design is tackled both for the state feedback and for the output feedback case in both cases the controller can be found solving a suitable set of lmis a typical engineering case study is included to illustrate the applicability of the devised conditions

a survey of numerical methods for ivps of odes with discontinuous right hand side this work is dedicated to the memory of donato trigiante who has been the first teacher of numerical analysis of the second author the authors remember donato as a generous teacher always ready to discuss with his students able to give them profound and interesting suggestions here we present a survey of numerical methods for differential systems with discontinuous right hand side in particular we will review methods where the discontinuities are detected by using an event function so called event driven methods and methods where the discontinuities are located by controlling the local errors so called time stepping methods particular attention will be devoted to discontinuous systems of filippov s type where sliding behavior on the discontinuity surface is allowed

output feedback decentralized control of large scale systems using weighted sensitivity functions minimization this paper considers the problem of achieving stability and desired dynamical transient behavior for linear large scale systems by decentralized control it can be done by making the effects of the interconnections between the subsystems arbitrarily small sufficient conditions for stability and diagonal dominance of the closed loop system are introduced these conditions are in terms of decentralized subsystems and directly make a constructive h control design possible a mixed h pole region placement is suggested such that by assigning the closed loop eigenvalues of the isolated subsystems appropriately the eigenvalues of the overall closed loop system are assigned in desirable range the designs are illustrated by an example

monotone methods for periodic solutions of second order scalar functional differential equations an iterative method is presented which starting from a lower or from an upper periodic solution provides a monotone sequence converging to a periodic solution of 1 with some restrictions on the growth off the method extends to functional differential equations of type 1 two numerical examples with an a posteriori error analysis are given

constraint aggregation in infinite dimensional spaces and applications an aggregation technique for constraints with values in hilbert spaces is suggested the technique allows us to replace the original optimization problem by a sequence of subproblems having scalar or finite dimensional constraints applications to optimal control games and stochastic programming are discussed in detail

numerical bifurcation analysis of delay differential equations numerical methods for the bifurcation analysis of delay differential equations ddes have only recently received much attention partially because the theory of ddes smoothness boundedness stability of solutions is more complicated and less established than the corresponding theory of ordinary differential equations as a consequence no established software packages exist at present for the bifurcation analysis of ddes we outline existing numerical methods for the computation and stability analysis of steady state solutions and periodic solutions of systems of ddes with several constant delays

output feedback decentralized stabilization ilmi approach in this paper the static output feedback decentralized stabilization problem is addressed using a linear matrix inequality approach a necessary and sufficient condition for static output feedback decentralized stabilizability is derived for linear time invariant large scale systems it is proven that the existence of a stabilizing decentralized gain is equivalent to that of the solution of a quadratic matrix inequality the extension of the result to h control is studied an iterative lmi algorithm based on the linear matrix inequality technique is proposed to obtain the decentralized feedback gain examples show the effectiveness of the algorithm

robust time optimal control of constrained linear systems a version of dynamic programming which computes level sets of the value function rather than the value function set itself is used to design robust non linear controllers for linear discrete time dynamical systems subject to hard constraints on controls and states the controller stabilizes the system and steers all trajectories emanating in a prescribed set to a control invariant set in minimum time for the robust regulator problem the control invariant terminal set is a neighborhood preferably small of the origin for the robust tracking problem the control invariant terminal set is a neighborhood of the invariant set in which the tracking error is zero two non linear controllers which utilize the level sets of the value function are described the first requires the controller to solve on line a modest linear program whose dimension is approximately the same as that of the control variable the second decomposes each level set into a set of simplices a piecewise linear control law affine in each simplex is then constructed

sliding motion in filippov differential systems theoretical results and a computational approach in this work we discuss some theoretical and numerical aspects of solving differential equations with discontinuous right hand sides of filippov type in particular i we propose second order corrections to the theory of filippov ii we provide a systematic and nonambiguous way to define the vector field on the intersection of several surfaces of discontinuity and iii we propose and implement a numerical method to approximate a trajectory of systems with discontinuous right hand sides and illustrate its performance on a few examples

an analysis of a class of neural networks for solving linear programming problems a class of neural networks that solve linear programming problems is analyzed the neural networks considered are modeled by dynamic gradient systems that are constructed using a parametric family of exact nondifferentiable penalty functions it is proved that for a given linear programming problem and sufficiently large penalty parameters any trajectory of the neural network converges in finite time to its solution set for the analysis lyapunov type theorems are developed for finite time convergence of nonsmooth sliding mode dynamic systems to invariant sets the results are illustrated via numerical simulation examples

the numerical solution of forward backward differential equations decomposition and related issues this paper focuses on the decomposition by numerical methods of solutions to mixed type functional differential equations mfdes into sums of forward solutions and backward solutions we consider equations of the form x t ax t bx t 1 cx t 1 and develop a numerical approach using a central difference approximation which leads to the desired decomposition and propagation of the solution we include illustrative examples to demonstrate the success of our method along with an indication of its current limitations

interpolation correction for collocation solutions of fredholm integro differential equations in this paper we discuss the collocation method for a large class of fredholm linear integro differential equations it will be shown that when a certain higher order interpolation operation is added to the collocation solution of this equation the new approximations will under suitable assumptions admit a multiterm error expansion in even powers of the step size h based on this expansion ideal multilevel correction results of this collocation solution are obtained

meromorphic lax representations of 1 1 dimensional multi hamiltonian dispersionless systems rational lax hierarchies introduced by krichever are generalized a systematic construction of infinite multi hamiltonian hierarchies and related conserved quantities is presented the method is based on the classical r matrix approach applied to poisson algebras a proof that poisson operators constructed near different points of laurent expansion of lax functions are equal is given all results are illustrated by several examples

brief an lmi condition for robust stability of polynomial matrix polytopes a sufficient lmi condition is proposed for checking robust stability of a polytope of polynomial matrices it hinges upon two recent results a new approach to polynomial matrix stability analysis and a new robust stability condition for convex polytopic uncertainty numerical experiments illustrate that the condition narrows significantly the unavoidable gap between conservative tractable quadratic stability results and exact np hard robust stability results

h 2 control for discrete time systems optimality and robustness abstract this paper proposes a new approach to determine h 2 optimal control for discrete time linear systems based on convex programming it is shown that all stabilizing state feedback control gains belong to a certain convex set well defined in a special parameter space the linear quadratic problem can be then formulated as the minimization of a linear objective over a convex set the optimal solution of this convex problem furnishes under certain conditions the same feedback control gain which is obtained from the classical discrete time riccati equation solution furthermore the method proposed can also handle additional constraints for instance the ones needed to assure asymptotical stability of discrete time systems under actuators failure some examples illustrate the theory

mixed type functional differential equations a numerical approach the equations considered in this paper are mixed type functional equations sometimes known as forward backward differential equations that take the form u t au t bu t 1 cu t 1 t i r we consider basic existence and uniqueness properties when i t 1 t 2 and we seek solutions u c t 1 1 t 2 1 that satisfy u t w 1 t t t 1 1 t 1 u t w 2 t t t 2 t 2 1 for prescribed functions w 1 w 2 absolutely continuous respectively on t 1 1 t 1 t 2 t 2 1 with arbitrary boundary conditions specified in this way the problem turns out to be ill posed and so existence and uniqueness questions have an important role to play in developing numerical schemes we consider with t 1 t 2 n numerical approximations of a solution when it exists the numerical methods that we consider are linear q methods and we investigate computationally their effectiveness through some illustrative examples

a comprehensive review of stability analysis of continuous time recurrent neural networks stability problems of continuous time recurrent neural networks have been extensively studied and many papers have been published in the literature the purpose of this paper is to provide a comprehensive review of the research on stability of continuous time recurrent neural networks including hopfield neural networks cohen grossberg neural networks and related models since time delay is inevitable in practice stability results of recurrent neural networks with different classes of time delays are reviewed in detail for the case of delay dependent stability the results on how to deal with the constant variable delay in recurrent neural networks are summarized the relationship among stability results in different forms such as algebraic inequality forms m matrix forms linear matrix inequality forms and lyapunov diagonal stability forms is discussed and compared some necessary and sufficient stability conditions for recurrent neural networks without time delays are also discussed concluding remarks and future directions of stability analysis of recurrent neural networks are given

a new gradient based neural network for solving linear and quadratic programming problems a new gradient based neural network is constructed on the basis of the duality theory optimization theory convex analysis theory lyapunov stability theory and lasalle invariance principle to solve linear and quadratic programming problems in particular a new function f x y is introduced into the energy function e x y such that the function e x y is convex and differentiable and the resulting network is more efficient this network involves all the relevant necessary and sufficient optimality conditions for convex quadratic programming problems for linear programming and quadratic programming qp problems with unique and infinite number of solutions we have proven strictly that for any initial point every trajectory of the neural network converges to an optimal solution of the qp and its dual problem the proposed network is different from the existing networks which use the penalty method or lagrange method and the inequality constraints are properly handled the simulation results show that the proposed neural network is feasible and efficient

the collocation method in the numerical solution of boundary value problems for neutral functional differential equations part i convergence results we consider the numerical solution of boundary value problems for general neutral functional differential equations by the collocation method the collocation method can be applied in two versions the finite element method and the spectral element method we give convergence results for the collocation method deduced by the convergence theory developed in s maset numer math 2015 pp 1 31 for a general discretization of an abstract reformulation of the problems such convergence results are then applied in part ii s maset siam j numer anal 53 2015 pp 2794 2821 of this paper to boundary values problems for a particular type of neutral functional differential equations namely differential equations with deviating arguments

quasi min max fuzzy model predictive control of direct methanol fuel cells abstract direct methanol fuel cells dmfcs are known as a promising power source in future in this paper we consider steering a dmfc plant to a desired operating point while optimizing the transient performance according to a quadratic cost function quasi min max fuzzy model predictive control fmpc with input constraints is proposed for the dmfc in order to reduce the computational burden for real time implementation a partial off line quasi min max fmpc is also proposed in this case a bank of invariant sets together with the corresponding feedback control laws are obtained by solving some linear matrix inequalities lmis off line leaving the online part a bisection search and a much simplified constrained optimization problem online computation complexity for both the quasi min max fmpc and the partial off line one is also analyzed simulation results are given to show the effectiveness of the proposed controllers

an abstract framework in the numerical solution of boundary value problems for neutral functional differential equations we consider the numerical solution of boundary value problems for general neutral functional differential equations the problems are restated in an abstract form and then a general discretization of the abstract form is introduced and a convergence analysis of this discretization is developed

convex optimization via feedbacks three dynamical systems are associated with a problem of convex optimization in a finite dimensional space for system trajectories x t the ratios x t t are respectively i solution tracking staying within the solution set x 0 ii solution abandoning reaching x 0 as time t goes back to the initial instant and iii solution approaching approaching x 0 as time t goes to infinity the systems represent a closed control system with appropriate feedbacks in typical cases the structure of the trajectories is simple enough for instance for a problem of quadratic programming with linear and box constraints solution approaching dynamics are described by a piecewise linear ode with a finite number of polyhedral domains of linearity finding the order of visiting these domains yields an analytic resolution of the original problem a detailed analysis is given for a particular example a discrete time approach is outlined

the algorithms for studying the trade offs between h attenuation and uncertainty tolerance uncertainty tolerance and h attenuations are two important concerns in feedback control design it is therefore important to develop effective methods to compute the trade off between these two objectives using a quasiconvex optimization approach this paper develops algorithms for two problems in this regard 1 given the uncertainty size of the system compute the minimum h norm bound of the closed loop system reachable by state feedback control and 2 given the required maximum acceptable h norm bound of the closed loop system compute the maximum uncertainty size which can be tolerated when state feedback control is applied a numerical example is included to show the trade off between uncertainty tolerance and h attenuations and the effectiveness of the algorithms

collocation methods for the computation of periodic solutions of delay differential equations in this paper we investigate collocation methods for the computation of periodic solutions of autonomous delay differential equations ddes periodic solutions are found by solving a periodic two point boundary value problem which is an infinite dimensional problem for ddes in contrast to the case of ordinary differential equations we investigate three collocation methods based on piecewise polynomials we discuss computational issues and show numerical orders of convergence using an extensive number of tests we compare our numerical results with known theoretical convergence results for initial value problems for ddes in particular we show how superconvergence at the mesh points can be lost or recovered depending on the dde model under consideration and on the choice of collocation discretization we end with a brief discussion of adaptive mesh selection

infinite dimensional frobenius manifolds for 2 1 integrable systems we introduce a structure of an infinite dimensional frobenius manifold on a subspace in the space of pairs of functions analytic inside outside the unit circle with simple poles at 0 respectively the dispersionless 2d toda equations are embedded into a bigger integrable hierarchy associated with this frobenius manifold