solving boundary value problems integral and integro differential equations using gegenbauer integration matrices we introduce a hybrid gegenbauer ultraspherical integration method hgim for solving boundary value problems bvps integral and integro differential equations the proposed approach recasts the original problems into their integral formulations which are then discretized into linear systems of algebraic equations using gegenbauer integration matrices gims the resulting linear systems are well conditioned and can be easily solved using standard linear system solvers a study on the error bounds of the proposed method is presented and the spectral convergence is proven for two point bvps tpbvps comparisons with other competitive methods in the recent literature are included the proposed method results in an efficient algorithm and spectral accuracy is verified using eight test examples addressing the aforementioned classes of problems the proposed method can be applied on a broad range of mathematical problems while producing highly accurate results the developed numerical scheme provides a viable alternative to other solution methods when high order approximations are required using only a relatively small number of solution nodes

new algorithms for unconstrained nonlinear optimal control problems a family of new first order algorithms for solving continuous time optimal control problems is presented the algorithms make use of the riccati matrix differential equation and are capable of solving the linear quadratic problem in one step the paper includes an analysis of the convergence of the proposed algorithms in the space of relaxed controls as well as the proof of the reduction of the cost functional at each iteration and numerical examples

the approximation for the boundary optimal control problem of burgers fisher equation with constraints this paper deals with the numerical approximation with meshless method for the boundary optimal control problem with some control and state constraints governed by the burgers fisher equation which is a nonlinear evolution equation and is the prototype model for the reaction convection and diffusion phenomena arising in many spatial temporal processes by making use of the element free galerkin efg method the original optimal control problem is discretized spatially to a semi discrete optimal control problem governed by a system of nonlinear ordinary differential equations then by using the control parameterization method the original problem can be reduced to an optimal parameter selection problem governed by a lumped parameter system which can be solved as a nonlinear optimization problems by using the sequential quadratic programming sqp algorithm the numerical simulations are given to illustrate the effectiveness of the proposed numerical approximation method

an optimization approach to state delay identification we consider a nonlinear delay differential system with unknown state delays our goal is to identify these state delays using experimental data to this end we formulate a dynamic optimization problem in which the state delays are decision variables and the cost function measures the discrepancy between predicted and observed system output we then show that the gradient of this problem s cost function can be computed by solving an auxiliary delay differential system by exploiting this result the state delay identification problem can be solved efficiently using a gradient based optimization method

dynamic optimization for robust path planning of horizontal oil wells this paper considers the three dimensional path planning problem for horizontal oil wells the decision variables in this problem are the curvature tool face angle and switching points for each turn segment in the path and the optimization objective is to minimize the path length and target error the optimal curvatures tool face angles and switching points can be readily determined using existing gradient based dynamic optimization techniques however in a real drilling process the actual curvatures and tool face angles will inevitably deviate from the planned optimal values thus causing an unexpected increase in the target error this is a critical challenge that must be overcome for successful practical implementation accordingly this paper introduces a sensitivity function that measures the rate of change in the target error with respect to the curvature and tool face angle of each turn segment based on the sensitivity function we propose a new optimization problem in which the switching points are adjusted to minimize target error sensitivity subject to continuous state inequality constraints arising from engineering specifications and an additional constraint specifying the maximum allowable increase in the path length from the optimal value our main result shows that the sensitivity function can be evaluated by solving a set of auxiliary dynamic systems by combining this result with the well known time scaling transformation we obtain an equivalent transformed problem that can be solved using standard nonlinear programming algorithms finally the paper concludes with a numerical example involving a practical path planning problem for a ci 16 cp146 well

convex control systems and convex optimal control problems with constraints this note discusses the concepts of convex control systems and convex optimal control problems we study control systems governed by ordinary differential equations in the presence of state and target constraints our note is devoted to the following main question under which additional assumptions is a sophisticated constrained optimal control problem equivalent to a simple convex minimization problem in a related hilbert space we determine some classes of convex control systems and show that for suitable cost functionals and constraints optimal control problems for these classes of systems correspond to convex optimization problems the latter can be reliably solved using standard numerical algorithms and effective regularization schemes in particular we propose a conceptual computational approach based on gradient type methods and proximal point techniques

a numerical comparison of chebyshev methods for solving fourth order semilinear initial boundary value problems in solving semilinear initial boundary value problems with prescribed non periodic boundary conditions using implicit explicit and implicit time stepping schemes both the function and derivatives of the function may need to be computed accurately at each time step to determine the best chebyshev collocation method to do this the accuracy of the real space chebyshev differentiation spectral space preconditioned chebyshev tau real space chebyshev integration and spectral space chebyshev integration methods are compared in the l 2 and w 2 2 norms when solving linear fourth order boundary value problems and in the l 0 t l 2 and l 0 t w 2 2 norms when solving initial boundary value problems we find that the best chebyshev method to use for high resolution computations of solutions to initial boundary value problems is the spectral space chebyshev integration method which uses sparse matrix operations and has a computational cost comparable to fourier spectral discretization

gmres for the differentiation operator we investigate using the gmres method with the differentiation operator this operator is unbounded and thus does not fall into the framework of existing krylov subspace theory we establish conditions under which a function can be approximated by its own derivatives in a domain of the complex plane these conditions are used to determine when gmres converges this algorithm outperforms traditional quadrature schemes for a large class of highly oscillatory integrals even when the kernel of oscillations is unknown

improved condition number for spectral methods introduction d une methode spectrale avec un nombre de conditionnement o n 2 n etant le degre maximum des polynomes

sqp methods for solving optimal control problems with control and state constraints adjoint variables sensitivity analysis and real time control parametric nonlinear optimal control problems subject to control and state constraints are studied two discretization methods are discussed that transcribe optimal control problems into nonlinear programming problems for which sqp methods provide efficient solution methods it is shown that sqp methods can be used also for a check of second order sufficient conditions and for a postoptimal calculation of adjoint variables in addition sqp methods lead to a robust computation of sensitivity differentials of optimal solutions with respect to perturbation parameters numerical sensitivity analysis is the basis for real time control approximations of perturbed solutions which are obtained by evaluating a first order taylor expansion with respect to the parameter the proposed numerical methods are illustrated by the optimal control of a low thrust satellite transfer to geosynchronous orbit and a complex control problem from aquanautics the examples illustrate the robustness accuracy and efficiency of the proposed numerical algorithms

theoretical analysis of a sinc nystrom method for volterra integro differential equations and its improvement abstract a sinc nystrom method for volterra integro differential equations was developed by zarebnia 2010 the method is quite efficient in the sense that exponential convergence can be obtained even if the given problem has endpoint singularity however its exponential convergence has not been proved theoretically in addition to implement the method the regularity of the solution is required although the solution is an unknown function in practice this paper reinforces the method by presenting two theoretical results 1 the regularity of the solution is analyzed and 2 its convergence rate is rigorously analyzed moreover this paper improves the method so that a much higher convergence rate can be attained and theoretical results similar to those listed above are provided numerical comparisons are also provided

an efficient spectral method for ordinary differential equations with rational function coefficients the authors present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients they employ expansions in terms of a large class of orthogonal polynomial families including all the classical orthogonal polynomials these families obey a simple 3 term recurrence relation for differentiation which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse the inverse is an integration operator for the family and it is simply the tridiagonal coefficient matrix for the recurrence since in these families convolution operators i e matrix representations of multiplication by a function are banded for polynomials the authors are able to obtain a banded representation for linear differential operators with rational coefficients this leads to a method of solution of initial or boundary value problems that besides having an operation count that scales linearly with the order of truncation n is computationally well conditioned among the applications considered is the use of rational maps for the resolution of sharp interior layers 19 refs 2 figs 3 tabs

a well conditioned collocation method using a pseudospectral integration matrix in this paper a well conditioned collocation method is constructed for solving general p th order linear differential equations with various types of boundary conditions based on a suitable birkhoff interpolation we obtain a new set of polynomial basis functions that results in a collocation scheme with two important features the condition number of the linear system is independent of the number of collocation points and the underlying boundary conditions are imposed exactly moreover the new basis leads to an exact inverse of the pseudospectral differentiation matrix of the highest derivative at interior collocation points which is therefore called the pseudospectral integration matrix psim we show that psim produces the optimal integration preconditioner and stable collocation solutions with even thousands of points

a note on the numerical solution of high order differential equations numerical solution of high order differential equations with multi boundary conditions is discussed in this paper motivated by the discrete singular convolution algorithm the use of fictitious points as additional unknowns is proposed in the implementation of locally supported lagrange polynomials the proposed method can be regarded as a local adaptive differential quadrature method two examples an eigenvalue problem and a boundary value problem which are governed by a sixth order differential equation and an eighth order differential equation respectively are employed to illustrate the proposed method

finding the zeros of a univariate equation proxy rootfinders chebyshev interpolation and the companion matrix when a function f x is holomorphic on an interval x in a b its roots on the interval can be computed by the following three step procedure first approximate f x on a b by a poly

spectral integration of linear boundary value problems spectral integration was deployed by orszag and co workers 1977 1980 1981 to obtain stable and efficient solvers for the incompressible navier stokes equation in rectangular geometries since then several variations of spectral integration have appeared in the literature in this article we derive yet more versions of spectral integration these new versions of spectral integration rely exclusively on banded matrices as opposed to banded matrices bordered with dense rows in addition we derive a factored form of spectral integration which relies only on bi and tri diagonal matrices key properties such as the accuracy of spectral integration even when green s functions are not resolved by the underlying grid and the accuracy of spectral integration in spite of ill conditioning of underlying linear systems are investigated timed comparisons show that reducing spectral integration to bi and tri diagonal systems leads to significant speed ups

gmres and integral operators in this paper we show how the properties of integral operators and their approximations are reflected in the performance of the gmres iteration and how these properties can be used to smooth the gmres iterates by an implicit application of nystrom interpolation thereby strengthening the norm in which convergence takes place the smoothed iteration has very similar properties to broyden s method we present an example to illustrate the ideas

brief paper optimal control problems with a continuous inequality constraint on the state and the control we consider an optimal control problem with a nonlinear continuous inequality constraint both the state and the control are allowed to appear explicitly in this constraint by discretizing the control space and applying a novel transformation a corresponding class of semi infinite programming problems is derived a solution of each problem in this class furnishes a suboptimal control for the original problem furthermore we show that such a solution can be computed efficiently using a penalty function method on the basis of these two ideas an algorithm that computes a sequence of suboptimal controls for the original problem is proposed our main result shows that the cost of these suboptimal controls converges to the minimum cost for illustration an example problem is solved

brief paper finite horizon dynamic optimization of nonlinear systems in real time a novel scheme for constructing and tracking the solution trajectories to regular finite horizon deterministic optimal control problems with nonlinear dynamics is devised the optimal control is obtained from the states and costates of hamiltonian odes integrated online in the one dimensional case the initial costate is found by successively solving two first order quasi linear partial differential equations whose independent variables are the time horizon duration t and the final penalty coefficient s these pdes should in general be integrated off line the solution rendering not only the missing initial condition sought in the particular t s situation but additional information on the boundary values of the whole two parameter family of control problems which can be used for designing the definitive objective functional optimal trajectories for the model are then generated in real time and used as references to be followed by the physical system numerical improvements are suggested for accurate integration of naturally unstable hamiltonian dynamics and strategies are proposed for tracking their results in finite time or asymptotically when perturbations in the state of the system appear the whole procedure is tested in models arising in aero navigation optimization

on the time complexity of worst case system identification in this paper we treat a general worst case system identification problem this problem is worst case with respect to both noise and system modeling uncertainty we consider this problem under various a priori information structures we determine bounds on the minimum duration identification experiment that must be run to identify the plant to within a specified guaranteed worst case error bound our results are algorithm independent we show that this minimum duration is prohibitively long based on our results we suggest that worst case with respect to noise system identification requires unrealistic amounts of experimental data

spectral integration and two point boundary value problems a numerical method for two point boundary value problems with constant coefficients is developed which is based on integral equations and the spectral integration matrix for chebyshev nodes the method is stable achieves superalgebraic convergence and requires o n log n operations where n is the number of nodes in the discretization although stable spectral methods have been constructed in the past they have generally been based on reformulating the recurrence relations obtained through spectral differentiation in an attempt to avoid the ill conditioning introduced by that process

high order methods for the numerical solution of volterra integro differential equations abstract if a first order volterra integro differential equation is solved by collocation in the space of continuous polynomial splines of degree m 1 with collocation occuring at the gauss legendre points then the resulting approximation u converges at its knots like o h 2m while its derivative u exhibits only o h m convergence this paper deals with the question of how to choose the collocation points so that both u and u converge like o h q with q maximal

computational method for a class of switched system optimal control problems we consider an optimal control problem with dynamics that switch between several subsystems of nonlinear differential equations each subsystem is assumed to satisfy a linear growth condition furthermore each subsystem switch is accompanied by an instantaneous change in the state these instantaneous changes called ldquostate jumpsrdquo are influenced by a set of control parameters that together with the subsystem switching times are decision variables to be selected optimally we show that an approximate solution for this optimal control problem can be computed by solving a sequence of conventional dynamic optimization problems existing optimization techniques can be used to solve each problem in this sequence a convergence result is also given to justify this approach

a survey of the maximum principles for optimal control problems with state constraints this paper gives a survey of the various forms of pontryagin s maximum principle for optimal control problems with state variable inequality constraints the relations between the different sets of optimality conditions arising in these forms are shown furthermore the application of these maximum principle conditions is demonstrated by solving some illustrative examples

optimal control of switched systems based on parameterization of the switching instants this paper presents a new approach for solving optimal control problems for switched systems we focus on problems in which a prespecified sequence of active subsystems is given for such problems we need to seek both the optimal switching instants and the optimal continuous inputs in order to search for the optimal switching instants the derivatives of the optimal cost with respect to the switching instants need to be known the most important contribution of the paper is a method which first transcribes an optimal control problem into an equivalent problem parameterized by the switching instants and then obtains the values of the derivatives based on the solution of a two point boundary value differential algebraic equation formed by the state costate stationarity equations the boundary and continuity conditions along with their differentiations this method is applied to general switched linear quadratic problems and an efficient method based on the solution of an initial value ordinary differential equation is developed an extension of the method is also applied to problems with internally forced switching examples are shown to illustrate the results in the paper

optimal control of switched systems and its parallel optimization algorithm in this paper a class of optimal switching control problems is considered in which the mode sequence of active subsystems and the number of mode switchings are not pre specified and both the switching sequence and the control inputs are to be chosen such that the cost functional is minimized for solving this problem we discuss the necessary conditions for optimality and the construction method for suboptimal solutions and develop a few sufficient conditions of judgment on the optimal or suboptimal solutions of the switched system according to the sufficient conditions a parallel computational algorithm is constructed to find optimal or suboptimal solutions for illustration two examples are solved using the proposed algorithm

control parametrization a unified approach to optimal control problems with general constraints abstract in this paper we consider a class of optimal control problems with general constraints the aim is to provide a unified approach to the numerical solution of this general class of optimal control problems by using the control parametrization technique where different types of constraints are shown to be equivalent to essentially the same functional form as the cost functional some convergence properties of this proposed method are also investigated

optimal control computation for nonlinear systems with state dependent stopping criteria in this paper we consider a challenging optimal control problem in which the terminal time is determined by a stopping criterion this stopping criterion is defined by a smooth surface in the state space when the state trajectory hits this surface the governing dynamic system stops by restricting the controls to piecewise constant functions we derive a finite dimensional approximation of the optimal control problem we then develop an efficient computational method based on nonlinear programming for solving the approximate problem we conclude the paper with four numerical examples

a combinatorial graph based solution method for a class of continuous time optimal control problems the paper addresses a class of continuous time optimal control problems whose solutions are typically characterized by both bang bang and singular control regimes analytical study and numerical computation of such solutions are very difficult and far from complete when only techniques from control theory are used this paper solves optimal control problems by reducing them to the combinatorial search for the shortest path in a specially constructed graph since the nodes of the graph are weighted in a sequence dependent manner we extend the classical shortest path algorithm to our case the proposed solution method is currently limited to single state problems with multiple control functions a production planning problem and a train operation problem are optimally solved to illustrate the method

fast accurate and small scale direct trajectory optimization using a gegenbauer transcription method this paper reports a novel direct gegenbauer ultraspherical transcription method gtm for solving continuous time optimal control oc problems ctocps with linear nonlinear dynamics and path constraints in elgindy et al 2012 1 we presented a gtm for solving nonlinear ctocps directly for the state and the control variables and the method was tailored to find the best path for an unmanned aerial vehicle mobilizing in a stationary risk environment this article extends the gtm to deal further with problems including higher order time derivatives of the states by solving the ctocp directly for the control u t and the highest order time derivative x n t n z the state vector and its derivatives up to the n 1 th order derivative can then be stably recovered by successive integration moreover we present our solution method for solving linear quadratic regulator lqr problems as we aim to cover a wider collection of ctocps with the concrete aim of comparing the efficiency of the current work with other classical discretization methods in the literature the proposed numerical scheme fully parameterizes the state and the control variables using gegenbauer expansion series for problems with various order time derivatives of the state variables arising in the cost function dynamical system or path terminal constraints the gtm seeks to fully parameterize the control variables and the highest order time derivatives of the state variables the time horizon is mapped onto the closed interval 0 1 the dynamical system characterized by differential equations is transformed into its integral formulation through direct integration the resulting problem on the finite interval is then transcribed into a nonlinear programming nlp problem through collocation at the gegenbauer gauss gg points the integral operations are approximated by optimal gegenbauer quadratures in a certain optimality sense the reduced nlp problem is solved in the gegenbauer spectral space and the state and the control variables are approximated on the entire finite horizon the proposed method achieves discrete solutions exhibiting exponential convergence using relatively small scale number of collocation points the advantages of the proposed direct gtm over other traditional discretization methods are shown through four well studied oc test examples the present work is a major breakthrough in the area of computational oc theory as it delivers significantly more accurate solutions using considerably smaller numbers of collocation points states and controls expansion terms moreover the gtm produces very small scale nlp problems which can be solved very quickly using the modern nlp software

how much uncertainty can be dealt with by feedback feedback is used primarily for reducing the effects of the plant uncertainty on the performance of control systems and as such understanding the following questions is of fundamental importance how much uncertainty can be dealt with by feedback what are the limitations of feedback how does the feedback performance depend quantitatively on the system uncertainty how can the capability of feedback be enhanced if a priori information about the system structure is available as a starting point toward answering these questions a typical class of first order discrete time dynamical control systems with both unknown nonlinear structure and unknown disturbances is selected for our investigation and some concrete answers are obtained in the paper in particular we find that in the space of unknown nonlinear functions the generalized lipschitz norm is a suitable measure for characterizing the size of the structure uncertainty and that the maximum uncertainty that can be dealt with by the feedback mechanism is described by a ball with radius 3 2 spl radic 2 in this normed function space

a new exact penalty method for semi infinite programming problems in this paper we consider a class of nonlinear semi infinite optimization problems these problems involve continuous inequality constraints that need to be satisfied at every point in an infinite index set as well as conventional equality and inequality constraints by introducing a novel penalty function to penalize constraint violations we form an approximate optimization problem in which the penalty function is minimized subject to only bound constraints we then show that this penalty function is exact that is when the penalty parameter is sufficiently large any local solution of the approximate problem can be used to generate a corresponding local solution of the original problem on this basis the original problem can be solved as a sequence of approximate nonlinear programming problems we conclude the paper with some numerical results demonstrating the applicability of our approach to pid control and filter design

iterative learning control design of nonlinear multiple time delay systems abstract most available results on iterative learning control address trajectory tracking problem for systems without multiple time delay this paper is concerned with iterative learning control design for nonlinear systems with multiple delays in which external disturbances and output measurement noises are involved we obtain some new and interesting criteria to guarantee the convergence of the tracking error in the sense of the norm it will be shown that the convergence of the system outputs to the desired trajectory is ensured in the absence of disturbances and output measurement noises in the presence of disturbance and measurement noises we estimate the upper bound of the tracking error in order to confirm the validity of the present results numerical simulation is also presented

runge kutta methods in optimal control and the transformed adjoint system the convergence rate is determined for runge kutta discretizations of nonlinear control problems the analysis utilizes a connection between the kuhn tucker multipliers for the discrete problem and the adjoint variables associated with the continuous minimum principle this connection can also be exploited in numerical solution techniques that require the gradient of the discrete cost function

optimal control of continuous time switched affine systems this paper deals with optimal control of switched piecewise affine autonomous systems where the objective is to minimize a performance index over an infinite time horizon we assume that the switching sequence has a finite length and that the decision variables are the switching instants and the sequence of operating modes we present two different approaches for solving such an optimal control problem the first approach iterates between a procedure that finds an optimal switching sequence of modes and a procedure that finds the optimal switching instants the second approach is inspired by dynamic programming and identifies the regions of the state space where an optimal mode switch should occur therefore providing a state feedback control law

optimal control of switching systems this paper considers an optimal control problem for a switching system for solving this problem we do not make any assumptions about the number of switches nor about the mode sequence they are determined by the solution of the problem the switching system is embedded into a larger family of systems and the optimization problem is formulated for the latter it is shown that the set of trajectories of the switching system is dense in the set of trajectories of the embedded system the relationship between the two sets of trajectories 1 motivates the shift of focus from the original problem to the more general one and 2 underlies the engineering relevance of the study of the second problem sufficient and necessary conditions for optimality are formulated for the second optimization problem if they exist bang bang type solutions of the embedded optimal control problem are solutions of the original problem otherwise suboptimal solutions are obtained via the chattering lemma

on minimum time paths of bounded curvature with position dependent constraints we consider the problem of a particle traveling from an initial configuration to a final configuration given by a point in the plane along with a prescribed velocity vector in minimum time with non homogeneous velocity and with constraints on the minimum turning radius of the particle over multiple regions of the state space necessary conditions for optimality of these paths are derived to characterize the nature of optimal paths both when the particle is inside a region and when it crosses boundaries between neighboring regions these conditions are used to characterize families of optimal and nonoptimal paths among the optimality conditions we derive a refraction law at the boundary of the regions that generalizes the so called snell s law of refraction in optics to the case of paths with bounded curvature tools employed to deduce our results include recent principles of optimality for hybrid systems a numerical example is given to demonstrate the derived results

brief paper initial condition of costate in linear optimal control using convex analysis the pontryagin maximum principle is one of the most important results in optimal control and provides necessary conditions for optimality in the form of a mixed initial terminal boundary condition on a pair of differential equations for the system state and its conjugate costate unfortunately this mixed boundary value problem is usually difficult to solve since the pontryagin maximum principle does not give any information on the initial value of the costate in this paper we explore an optimal control problem with linear and convex structure and derive the associated dual optimization problem using convex duality which is often much easier to solve than the original optimal control problem we present that the solution to the dual optimization problem supplements the necessary conditions of the pontryagin maximum principle and elaborate the procedure of constructing the optimal control and its corresponding state trajectory in terms of the solution to the dual problem

distributed optimal control for multi agent trajectory optimization this paper presents a novel optimal control problem referred to as distributed optimal control that is applicable to multiscale dynamical systems comprised of numerous interacting agents the system performance is represented by an integral cost function of the macroscopic state that is optimized subject to a hyperbolic partial differential equation known as the advection equation the microscopic control laws are derived from the optimal macroscopic description using a potential function approach the optimality conditions of the distributed optimal control problem are first derived analytically and then demonstrated numerically through a multi agent trajectory optimization problem

on metric dimensions of discrete time systems in this paper metric dimensions of a class of discrete time systems are derived the n width and e dimension of the class in l1 kernel norm are obtained and a corresponding minimal e net is constructed by time domain truncation of system kernels asymptotic estimates of the n width and e dimension of the class in h norm are established

extension of chebfun to periodic functions algorithms and underlying mathematics are presented for numerical computation with periodic functions via approximations to machine precision by trigonometric polynomials including the solution of linear and nonlinear periodic ordinary differential equations differences from the nonperiodic chebyshev case are highlighted

fast identification n widths and uncertainty principles for lti and slowly varying systems the optimal worst case uncertainty that can be achieved by identification depends on the observation time in the first part of the paper this dependence is evaluated for selected linear time invariant systems in the l sup 1 and h sup spl infin norms and shown to be derivable from a monotonicity principle the minimal time required is shown to depend on the metric complexity of the a priori information set two notions of n width or metric dimension are introduced to characterize this complexity in the second part of the paper the results are applied to systems in which the law governing the evolution of the uncertain elements is not time invariant such systems cannot be identified accurately the inherent uncertainty is bounded in the case of slow time variation the n widths and related optimal inputs provide benchmarks for the evaluation of actual inputs occurring in adaptive feedback systems

time optimal zermelo s navigation problem with moving and fixed obstacles in this paper we consider a time optimal zermelo s navigation problem znp with moving and fixed obstacles this problem can be formulated as an optimal control problem with continuous inequality constraints and terminal state constraints by using the control parametrization technique together with the time scaling transform the problem is transformed into a sequence of optimal parameters selection problems with continuous inequality constraints and terminal state constraints for each problem an exact penalty function method is used to append all the constraints to the objective function yielding a new unconstrained optimal parameters selection problem it is solved as a nonlinear optimization problem different scenarios are considered in the simulation and the results obtained show that the proposed method is effective

optimal control of a class of hybrid systems we present a modeling framework for hybrid systems intended to capture the interaction of event driven and time driven dynamics this is motivated by the structure of many manufacturing environments where discrete entities termed jobs are processed through a network of workcenters so as to change their physical characteristics associated with each job is a temporal state subject to event driven dynamics and a physical state subject to time driven dynamics based on this framework we formulate and analyze a class of optimal control problems for single stage processes first order optimality conditions are derived and several properties of optimal state trajectories sample paths are identified which significantly simplify the task of obtaining explicit optimal control policies

brief paper optimal control problems with multiple characteristic time points in the objective and constraints in this paper we develop a computational method for a class of optimal control problems where the objective and constraint functionals depend on two or more discrete time points these time points can be either fixed or variable using the control parametrization technique and a time scaling transformation this type of optimal control problem is approximated by a sequence of approximate optimal parameter selection problems each of these approximate problems can be viewed as a finite dimensional optimization problem new gradient formulae for the cost and constraint functions are derived with these gradient formulae standard gradient based optimization methods can be applied to solve each approximate optimal parameter selection problem for illustration two numerical examples are solved

optimal control of switched systems via non linear optimization based on direct differentiations of value functions this paper presents an approach for solving optimal control problems of switched systems in general in such problems one needs to find both optimal continuous inputs and optimal switching sequences since the system dynamics vary before and after every switching instant after formulating a general optimal control problem we propose a two stage optimization methodology since many practical problems only concern optimization where the number of switchings and the sequence of active subsystems are given we concentrate on such problems and propose a method which uses nonlinear optimization and is based on direct differentiations of value functions the method is then applied to general switched linear quadratic gslq problems examples illustrate the results

method of evolving junctions a new approach to optimal control with constraints abstract we propose a new strategy called method of evolving junctions mej to compute the solutions for a class of optimal control problems with constraints on both state and control variables our main idea is that by leveraging the geometric structures of the optimal solutions we recast the infinite dimensional optimal control problem into an optimization problem depending on a finite number of points called junctions then using a modified gradient flow method whose dimension can change dynamically we find local solutions for the optimal control problem we also employ intermittent diffusion a global optimization method based on stochastic differential equations to obtain the global optimal solution we demonstrate via a numerical example that mej can effectively solve path planning problems in dynamical environments

vibrations of non uniform rings studied by means of the differential quadrature method abstract the vibrational behavior of ring like structures is of interest in several technological situations from automotive and aircraft tires to electric motors and transformers this study deals with the solution of the governing sixth order differential equation by using the differential quadrature method the fundamental frequency of in plane inextensional vibration is thus determined and compared with the value predicted by the optimized rayleigh ritz method

optimal boundary control for water hammer suppression in fluid transmission pipelines when fluid flow in a pipeline is suddenly halted a pressure surge or wave is created within the pipeline this phenomenon called water hammer can cause major damage to pipelines including pipeline ruptures in this paper we model the problem of mitigating water hammer during valve closure by an optimal boundary control problem involving a nonlinear hyperbolic pde system that describes the fluid flow along the pipeline the control variable in this system represents the valve boundary actuation implemented at the pipeline terminus to solve the boundary control problem we first use the method of lines to obtain a finite dimensional ode model based on the original pde system then for the boundary control design we apply the control parameterization method to obtain an approximate optimal parameter selection problem that can be solved using nonlinear optimization techniques such as sequential quadratic programming sqp we conclude the paper with simulation results demonstrating the capability of optimal boundary control to significantly reduce flow fluctuation

automatic spectral collocation for integral integro differential and integrally reformulated differential equations automatic chebyshev spectral collocation methods for fredholm and volterra integral and integro differential equations have been implemented as part of the chebfun software system this system enables a symbolic syntax to be applied to numerical objects in order to pose and solve problems without explicit references to discretization the same objects can be used in matrix free iterative methods in linear algebra in order to avoid very large dense matrices or allow application to problems with nonsmooth coefficients as a further application of the ability to implement operator equations a method of greengard 1 for the recasting of differential equations as integral equations is generalized to mth order boundary value and generalized eigenvalue problems in the integral form large condition numbers associated with differentiation matrices in high order problems are avoided the ability to implement the recasting process generally follows from implementation of the operator expressions in chebfun the integral method also can be extended to first order systems although chebfun syntax does not currently allow easy implementation in this case

numerical solution of a fredholm integro differential equation modelling θ neural networks we propose several approaches to the numerical solution of a new fredholm integro differential equations modelling neural networks a solution strategy based on expansions onto standard cardinal basis functions and collocation is presented comparative numerical experiments illustrate specific advantages and drawbacks of the different approaches and are used to motivate alternate strategies

controller design and value function approximation for nonlinear dynamical systems this work considers the infinite time discounted optimal control problem for continuous time input affine polynomial dynamical systems subject to polynomial state and box input constraints we propose a sequence of sum of squares sos approximations of this problem obtained by first lifting the original problem into the space of measures with continuous densities and then restricting these densities to polynomials these approximations are tightenings rather than relaxations of the original problem and provide a sequence of rational controllers with value functions associated to these controllers converging under some technical assumptions to the value function of the original problem in addition we describe a method to obtain polynomial approximations from above and from below to the value function of the extracted rational controllers and a method to obtain approximations from below to the optimal value function of the original problem thereby obtaining a sequence of asymptotically optimal rational controllers with explicit estimates of suboptimality numerical examples demonstrate the approach

stabilization for a class of second order switched systems abstract this paper considers the asymptotic stabilization problem of second order linear time invariant lti autonomous switched systems consisting of two subsystems with unstable focus equilibrium more precisely we find a necessary and sufficient condition for the origin to be asymptotically stable under the predesigned switching law the result is obtained without looking for a common lyapunov function or multiple lyapunov function but studying the locus in which the two subsystem s vector fields are parallel then the most stabilizing switching laws are designed which have translated the switched system into a piecewise linear system two numerical examples are presented to show the potential of the proposed techniques

information based complexity of uncertainty sets in feedback control a notion of information based complexity is introduced to characterize complexities of plant uncertainty sets in feedback control settings and to understand relationships between identification and feedback control in dealing with uncertainty this new complexity measure extends the kolmogorov entropy to problems involving information acquisition identification and processing control and provides a tangible measure of difficulty of an uncertainty set of plants in the special cases of robust stabilization for systems with either gain uncertainty or unstructured additive uncertainty the complexity measures are explicitly derived

a unified framework for hybrid control model and optimal control theory we propose a very general framework that systematizes the notion of a hybrid system combining differential equations and automata governed by a hybrid controller that issues continuous variable commands and makes logical decisions we first identify the phenomena that arise in real world hybrid systems then we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems evolving on continuous variable state spaces and subject to continuous controls and discrete transitions the model captures the identified phenomena subsumes previous models yet retains enough structure to pose and solve meaningful control problems we develop a theory for synthesizing hybrid controllers for hybrid plants in all optimal control framework in particular we demonstrate the existence of optimal relaxed and near optimal precise controls and derive generalized quasi variational inequalities that the associated value function satisfies we summarize algorithms for solving these inequalities based on a generalized bellman equation impulse control and linear programming

optimal shortcuts to adiabaticity for a quantum piston abstract in this paper we use optimal control to design minimum time adiabatic like paths for the expansion of a quantum piston under realistic experimental constraints we calculate the minimum expansion time and compare it with that obtained from a state of the art inverse engineering method we use this result to rederive the known upper bound for the cooling rate of a refrigerator which provides a quantitative description for the unattainability of absolute zero the third law of thermodynamics we finally point out the relation of the present work to the fast adiabatic like expansion of an accordion optical lattice a system which can be used to magnify the initial quantum state quantum microscope

robust suboptimal control of nonlinear systems in this paper we consider a nonlinear dynamic system with uncertain parameters our goal is to choose a control function for this system that balances two competing objectives i the system should operate efficiently and ii the system s performance should be robust with respect to changes in the uncertain parameters with this in mind we introduce an optimal control problem with a cost function penalizing both the system cost a function of the final state reached by the system and the system sensitivity the derivative of the system cost with respect to the uncertain parameters we then show that the system sensitivity can be computed by solving an auxiliary initial value problem this result allows one to convert the optimal control problem into a standard mayer problem which can be solved directly using conventional techniques we illustrate this approach by solving two example problems using the software miser3

the sample complexity of worst case identification of fir linear systems we consider the problem of identification of linear systems in the presence of measurement noise which is unknown but bounded in magnitude by some 0 we focus on the case of linear systems with a finite impulse response it is known that the optimal identification error is related within a factor of 2 to the diameter of a so called uncertainty set and that the latter diameter is upper bounded by 2 if a sufficiently long identification experiment is performed we establish that for any k 1 the minimal length of an identification experiment that is guaranteed to lead to a diameter bounded by 2k behaves like 2nf 1k when n is large where n is the length of the impulse response and is a positive function known in closed form while the framework is entirely deterministic our results are proved using probabilistic tools

an extension of matlab to continuous functions and operators an object oriented matlab system is described for performing numerical linear algebra on continuous functions and operators rather than the usual discrete vectors and matrices about eighty matlab functions from plot and sum to svd and cond have been overloaded so that one can work with our chebfun objects using almost exactly the usual matlab syntax all functions live on 1 1 and are represented by values at sufficiently many chebyshev points for the polynomial interpolant to be accurate to close to machine precision each of our overloaded operations raises questions about the proper generalization of familiar notions to the continuous context and about appropriate methods of interpolation differentiation integration zerofinding or transforms applications in approximation theory and numerical analysis are explored and possible extensions for more substantial problems of scientific computing are mentioned

a spectral collocation method based on integrated chebyshev polynomials for two dimensional biharmonic boundary value problems this paper reports a new spectral collocation method for numerically solving two dimensional biharmonic boundary value problems the construction of the chebyshev approximations is based on integration rather than conventional differentiation this use of integration allows i the imposition of the governing equation at the whole set of grid points including the boundary points and ii the straightforward implementation of multiple boundary conditions the performance of the proposed method is investigated by considering several biharmonic problems of first and second kinds more accurate results and higher convergence rates are achieved than with conventional differential methods

computation of frequency responses for linear time invariant pdes on a compact interval we develop mathematical framework and computational tools for calculating frequency responses of linear time invariant pdes in which an independent spatial variable belongs to a compact interval in conventional studies this computation is done numerically using spatial discretization of differential operators in the evolution equation in this paper we introduce an alternative method that avoids the need for finite dimensional approximation of the underlying operators in the evolution model this method recasts the frequency response operator as a two point boundary value problem and uses state of the art automatic spectral collocation techniques for solving integral representations of the resulting boundary value problems with accuracy comparable to machine precision our approach has two advantages over currently available schemes first it avoids numerical instabilities encountered in systems with differential operators of high order and second it alleviates difficulty in implementing boundary conditions we provide examples from newtonian and viscoelastic fluid dynamics to illustrate utility of the proposed method

an effective pseudospectral method for constraint dynamic optimisation problems with characteristic times dynamic optimisation problem with characteristic times widely existing in many areas is one of the frontiers and hotspots of dynamic optimisation researches this paper considers a class of dynam

a computational method for solving time delay optimal control problems with free terminal time abstract this paper considers a class of optimal control problems for general nonlinear time delay systems with free terminal time we first show that for this class of problems the well known time scaling transformation for mapping the free time horizon into a fixed time interval yields a new time delay system in which the time delays are variable then we introduce a control parameterization scheme to approximate the control variables in the new system by piecewise constant functions this yields an approximate finite dimensional optimization problem with three types of decision variables the control heights the control switching times and the terminal time in the original system which influences the variable time delays in the new system we develop a gradient based optimization approach for solving this approximate problem simulation results are also provided to demonstrate the effectiveness of the proposed approach

brief paper optimal switching instants for a switched capacitor dc dc power converter we consider a switched capacitor dc dc power converter with variable switching instants the determination of optimal switching instants giving low output ripple and strong load regulation is posed as a non smooth dynamic optimization problem by introducing a set of auxiliary differential equations and applying a time scaling transformation we formulate an equivalent optimization problem with semi infinite constraints existing algorithms can be applied to solve this smooth semi infinite optimization problem the existence of an optimal solution is also established for illustration the optimal switching instants for a practical switched capacitor dc dc power converter are determined using this approach

a legendre pseudospectral method for computing travelling waves with corners slope discontinuities in one space dimension with application to whitham s equation family if the dispersion in a nonlinear hyperbolic wave equation is weak in the sense that the frequency k of cos kx is bounded as k it is common that i travelling waves exist up to a limiting amplitude with wave breaking for higher amplitudes and ii the limiting wave has a corner that is a discontinuity in slope because corner waves are not smooth standard numerical methods converge poorly as the number of grid points is increased however the corner wave is important because at least in some systems it is the attractor for all large amplitude initial conditions here we devise a legendre pseudospectral method which is uncorrupted by the singularity the symmetric u x u x wave can be computed on an interval spanning only half the spatial period since u is smooth on this domain which does not include the corner except as an endpoint all numerical difficulties are avoided a key step is to derive an extra boundary condition which uniquely identifies the corner wave with both the grid point values of u x and phase speed c as unknowns the discretized equations imposing three boundary conditions on a second order differential equation are solved by a newton raphson iteration although our method is illustrated by the so called whitham s equation ut uux du dx where d is a very general linear operator the ideas are widely applicable

optimal gegenbauer quadrature over arbitrary integration nodes this paper treats definite integrations numerically using gegenbauer quadratures the novel numerical scheme introduces the idea of exploiting the strengths of the chebyshev legendre and gegenbauer polynomials through a unified approach and using a unique numerical quadrature in particular the developed numerical scheme employs the gegenbauer polynomials to achieve rapid rates of convergence of the quadrature for the small range of the spectral expansion terms for a large scale number of expansion terms the numerical quadrature has the advantage of converging to the optimal chebyshev and legendre quadratures in the l norm and l 2 norm respectively the key idea is to construct the gegenbauer quadrature through discretizations at some optimal sets of points of the gegenbauer gauss gg type in a certain optimality sense we show that the gegenbauer polynomial expansions can produce higher order approximations to the definite integrals 1 x if x dx of a smooth function f x c 1 1 for the small range by minimizing the quadrature error at each integration point x i through a pointwise approach the developed gegenbauer quadrature can be applied for approximating integrals with any arbitrary sets of integration nodes exact integrations are obtained for polynomials of any arbitrary degree n if the number of columns in the developed gegenbauer integration matrix gim is greater than or equal to n the error formula for the gegenbauer quadrature is derived moreover a study on the error bounds and the convergence rate shows that the optimal gegenbauer quadrature exhibits very rapid convergence rates faster than any finite power of the number of gegenbauer expansion terms two efficient computational algorithms are presented for optimally constructing the gegenbauer quadrature we illustrate the high order approximations of the optimal gegenbauer quadrature through extensive numerical experiments including comparisons with conventional chebyshev legendre and gegenbauer polynomial expansion methods the present method is broadly applicable and represents a strong addition to the arsenal of numerical quadrature methods

non euclidian metrics and the robust stabilization of systems with parameter uncertainty this paper considers from a complex function theoretic point of view certain kinds of robust synthesis problems in particular we use a certain kind of metric on the disk the hyperbolic metric which allows us to reduce the problem of robust stabilization of systems with many types of real and complex parameter variations to an easily solvable problem in non euclidian geometry it is shown that several apparently different problems can be treated in a unified general framework a new result on the gain margin problem for multivariable plants is also given finally we apply our methods to systems with real zero or pole variations