l2 analytic torsion abstract in this paper we introduce a new differential invariant called l2 analytic torsion for closed manifolds with positive decay and whose universal covers have trivial l2 cohomology l2 analytic torsion can be thought of as a suitable generalization of the ray singer analytic torsion we establish various functorial properties of l2 analytic torsion and also compute it for odd dimensional closed hyperbolic manifolds with the help of results from fried

disturbance attenuation for constrained discrete time systems via receding horizon controls in this note we propose new receding horizon h sub spl infin control rhhc schemes for linear input constrained discrete time invariant systems with disturbances the proposed control schemes are based on the dynamic game problem of a finite horizon cost function with a fixed finite terminal weighting matrix and a one horizon cost function with time varying finite terminal weighting matrices respectively we show that the resulting rhhcs guarantee closed loop stability in the absence of disturbances and h sub spl infin norm bound for 2 norm bounded disturbances we also show that the proposed schemes can easily be implemented via linear matrix inequality optimization we illustrate the effectiveness of the proposed schemes through simulations

output integral sliding mode for min max optimization of multi plant linear uncertain systems we consider the application of a min max optimal control based on the lq index for a set of systems where only the output information is available here every system is affected by matched uncertainties and we propose to use an output integral sliding mode to compensate the matched uncertainties right after the beginning of the process for the case when the extended system is free of invariant zeros a hierarchical sliding mode observer is applied the error of realization of the proposed control algorithm is estimated in terms of the sampling step and actuator time constant an example illustrates the suggested method of design

cheeger muller theorem on manifolds with cusps we prove equality between the renormalized ray singer analytic torsion and the intersection r torsion on a witt manifold with cusps up to an error term determined explicitly by the betti numbers of the cross section of the cusp and the intersection r torsion of a model cone in the first step of the proof we compute explicitly the renormalized ray singer analytic torsion of a model cusp in general dimension and without the witt condition in the second step we establish a gluing formula for renormalized ray singer analytic torsion on a general class of non compact manifolds in any dimension that includes witt manifolds with cusps but also scattering manifolds with asymptotically conical ends in the final step a cheeger muller theorem on cusps follows by a combination of the previous explicit computation and the gluing formula

application of shifted jacobi pseudospectral method for solving in finite horizon min max optimal control problems with uncertainty abstractthe difficulty of solving the min max optimal control problems m mocps with uncertainty using generalised euler lagrange equations is caused by the combination of split boundary conditions nonlinear differential equations and the manner in which the final time is treated in this investigation the shifted jacobi pseudospectral method sjpm as a numerical technique for solving two point boundary value problems tpbvps in m mocps for several boundary states is proposed at first a novel framework of approximate solutions which satisfied the split boundary conditions automatically for various boundary states is presented then by applying the generalised euler lagrange equations and expanding the required approximate solutions as elements of shifted jacobi polynomials finding a solution of tpbvps in nonlinear m mocps with uncertainty is reduced to the solution of a system of algebraic equations moreover the jacobi polynomials are particularly useful for boundary value problems in unbounded

mini max integral sliding mode control for multimodel linear uncertain systems an original linear time varying system with matched and unmatched disturbances and uncertainties is replaced by a finite set of dynamic models such that each one describes a particular uncertain case including exact realizations of possible dynamic equations as well as external unmatched bounded disturbances such a tradeoff between an original uncertain linear time varying dynamic system and a corresponding higher order multimodel system containing only matched uncertainties leads to a linear multi model system with known unmatched bounded disturbances and unknown matched disturbances as well each model from a given finite set is characterized by a quadratic performance index the developed minimax integral sliding mode control strategy gives an optimal minimax linear quadratic lq control with additional integral sliding mode term the design of this controller is reduced to a solution of an equivalent mini max lq problem that corresponds to the weighted performance indices with weights from a finite dimensional simplex the additional integral sliding mode controller part completely dismisses the influence of matched uncertainties from the initial time instant two numerical examples illustrate this study

the dynamic programming approach to multi model robust optimization abstract the aim of this paper is to extend the dynamic programming dp approach to multi model optimal control problems ocps we deal with robust optimization of multi model control systems and are particularly interested in the hamilton jacobi bellman hjb equation for the above class of problems in this paper we study a variant of the hjb for multi model ocps and examine the natural relationship between the bellman dp techniques and the robust maximum principle mp moreover we describe how to carry out the practical calculations in the context of multi model lq problems and derive the associated riccati type equation

optimum performance levels for minimax filters predictors and smoothers abstract for both discrete and continuous time linear time varying systems we obtain the achievable performance levels for minimax filters predictors and smoothers in terms of the finite escape times of some related discrete and continuous time riccati equations our game theoretic approach also yields an alternative derivation for the corresponding minimax estimators which were first obtained in 1 they are all bayes estimators with respect to particular gaussian distributions and admit recursive structures

constrained stochastic lqc a tractable approach despite the celebrated success of dynamic programming for optimizing quadratic cost functions over linear systems such an approach is limited by its inability to tractably deal with even simple constraints in this paper we present an alternative approach based on results from robust optimization to solve the stochastic linear quadratic control slqc problem in the unconstrained case the problem may be formulated as a semidefinite optimization problem sdp we show that we can reduce this sdp to optimization of a convex function over a scalar variable followed by matrix multiplication in the current state thus yielding an approach that is amenable to closed loop control and analogous to the riccati equation in our framework we also consider a tight second order cone socp approximation to the sdp that can be solved much more efficiently when the problem has additional constraints both the sdp and socp are tractable in the presence of control and state space constraints moreover compared to the riccati approach they provide much greater control over the stochastic behavior of the cost function when the noise in the system is distributed normally

on torsion in the cohomology of locally symmetric varieties the main result of this paper is the existence of galois representations associated with the mod p or mod p m cohomology of the locally symmetric spaces for gln over a totally real or cm eld proving conjectures of ash and others following an old suggestion of clozel recently realized by harris lan taylor thorne for characteristic 0 cohomology classes one realizes the cohomology of the locally symmetric spaces for gln as a boundary contribution of the cohomology of symplectic or unitary shimura varieties so that the key problem is to understand torsion in the cohomology of shimura varieties thus we prove new results on the p adic geometry of shimura varieties of hodge type namely the shimura varieties become perfectoid when passing to the inverse limit over all levels at p and a new period map towards the ag variety exists on them called the hodge tate period map it is roughly analogous to the embedding of the hermitian symmetric domain which is roughly the inverse limit over all levels of the complex points of the shimura variety into its compact dual the hodge tate period map has several favorable properties the most important being that it commutes with the hecke operators away from p for the trivial action of these hecke operators on the ag variety and that automorphic vector bundles come via pullback from the ag variety

analytic torsion of complete hyperbolic manifolds of finite volume abstract in this paper we define the analytic torsion for a complete oriented hyperbolic manifold of finite volume it depends on a representation of the fundamental group for manifolds of odd dimension we study the asymptotic behavior of the analytic torsion with respect to certain sequences of representations obtained by restriction of irreducible representations of the group of isometries of the hyperbolic space to the fundamental group

development of a block lanczos algorithm for free vibration analysis of spinning structures this paper is concerned with the development of an efficient eigenproblem solution algorithm and an associated computer program for the economical solution of the free vibration problem of complex practical spinning structural systems thus a detailed description of a newly developed block lanczos procedure is presented in this paper that employs only real numbers in all relevant computations and also fully exploits sparsity of associated matrices the procedure is capable of computing multiple roots and proves to be most efficient compared to other existing similar techniques

minimax optimal control this paper provides a framework for deriving necessary conditions in the form of a maximum principle for minimax optimal control problems the distinguishing feature of these problems is that the data depends on a vector alpha of unknown parameters and optimality is defined on a worst case basis as alpha ranges over the parameter set cal a the centerpiece a minimax maximum principle is a set of optimality conditions for such problems here the parameter set cal a is taken to be an arbitrary compact metric space and the hypotheses imposed on the dynamics and endpoint constraints are of an unrestrictive nature the minimax maximum principle captures as special cases necessary conditions for optimal control problems with minimax costs for problems involving semi infinite endpoint constraints and also a maximum principle for state constrained optimal control problems

heat kernels on covering spaces and topological invariants it is well known that there are relationships between the heat flow acting on differential forms on a closed oriented manifold m and the topology of m from hodge theory one can recover the betti numbers of m from the heat flow furthermore ray and singer 42 defined an analytic torsion a smooth invariant of acyclic flat bundles on m and conjectured that it equals the classical reidemeister torsion this conjecture was proved to be true independently by cheeger 7 and muller 40 the analytic torsion is nonzero only on odd dimensional manifolds and behaves in some ways as an odd dimensional counterpart of the euler characteristic 20 if m is not simply connected then there is a covering space analog of the betti numbers using the heat flow on the universal cover m one can define the l2 betti numbers of m 1 by taking the trace of the heat kernel not in the ordinary sense but as an element of a certain type ii von neumann algebra more concretely this amounts to integrating the local trace of the heat kernel over a fundamental domain in m in 11 we summarize this theory we consider the covering space analog of the analytic torsion this i analytic torsion has the same relation to the i cohomology as the ordinary analytic torsion bears to de rham cohomology in 111 we define the i analytic torsion 9 m under a technical assumption which we discuss later and prove its basic properties we show that m is a smooth invariant of manifolds whose l2 betti numbers vanish the proof is similar to that of the analogous statement for the ordinary analytic torsion but requires some care because of the possible slow decay of the heat kernels for large time in order to know if there are interesting examples of m we compute it in the case where m admits a hyperbolic metric it is clear that is proportionate to the volume of the hyperbolic metric which is

robust stochastic maximum principle for multi model worst case optimization this paper develops a version of the robust maximum principle applied to the minimax mayer problem formulated for stochastic differential equations with a control dependent diffusion term the parametric families of first and second order adjoint stochastic processes are introduced to construct the corresponding hamiltonian formalism the hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic hamiltonians corresponding to each possible value of the parameter the cost function is defined on a finite horizon and contains the mathematical expectation of a terminal term a terminal condition given by a vector function is also considered the optimal control strategies adapted for available information for the wide class of multi model systems given by a stochastic differential equation with parameters from a given finite set are constructed this problem belongs to the class of minimax stochastic optimization problems the p

flexural vibration of rotating rectangular plates of variable thickness abstract free vibration of rotating rectangular plates with an exponentially varying thickness is studied the plate is clamped at the rim of a central hub and is free along the other three edges in the general formulation the plate is considered to be made of orthotropic materials the differential equations governing the in plane displacements before vibration takes place are solved by a finite difference method the curve fitted previbratory in plane stress distribution based on the numerical results is used in the equation for the flexural vibration of the plate and the natural frequencies are subsequently determined by the extended galerkin procedure some numerical results are presented for illustrative purposes

free vibration behavior of spinning shear deformable plates composedof composite materials the theory accounts for geometric nonlinearity in the form of the von karman strains and the effects of rotatory inertia the plate is permitted to have an arbitrary orientation offset from the axis of rotation a finite element model is developed the mode shapes as functions of angular velocity pitch angle and sweep angle are presented

a solution of the eigenproblem for undamped gyroscopic systems with the lanczos algorithm this paper presents an efficient numerical solution of the quadratic eigenproblem arising in the analysis of gyroscopic systems such problems are known to reduce to a generalized linear eigenproblem defined by two real non singular matrices one symmetric and one skew symmetric for this class of problems the general lanczos algorithm for unsymmetric matrices is shown to simplify considerably and yields an efficient solution of the problem full advantage can be taken of the sparsity of the matrices and of the specific nature of gyroscopic systems numerical examples are presented which demonstrate the efficiency and accuracy of the solution procedure

robust dynamic programming for min max model predictive control of constrained uncertain systems we address min max model predictive control mpc for uncertain discrete time systems by a robust dynamic programming approach and develop an algorithm that is suitable for linearly constrained polytopic systems with piecewise affine cost functions the method uses polyhedral representations of the cost to go functions and feasible sets and performs multiparametric programming by a duality based approach in each recursion step we show how to apply the method to robust mpc and give conditions guaranteeing closed loop stability finally we apply the method to a tutorial example a parking car with uncertain mass

robust optimal control for minimax stochastic linear quadratic problem the robust maximum principle applied to the minimax linear quadratic problem is derived for stochastic differential equations containing a control dependent diffusion term the parametric families of the first and second order adjoint stochastic processes are obtained to construct the corresponding hamiltonian formalism the hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic hamiltonians corresponding to each value of the uncertain parameter from a given finite set the cost function is considered on a finite horizon contains the mathematical expectation of both an integral and a terminal term and on an infinite one a time averaged losses function these problems belong to the class of minimax stochastic optimization problems it is shown that the construction of the minimax optimal controller can be reduced to an optimization problem on a finitedimensional simplex and consists in the analysis of the dependence of ric

the torsion in symmetric powers on congruence subgroups of bianchi groups in this paper we prove that for a fixed neat principal congruence subgroup of a bianchi group the order of the torsion part of its second cohomology group with coefficients in an integral lattice associated to the m th symmetric power of the standard representation of sl 2 c grows exponentially in m 2 we give upper and lower bounds for the growth rate our result extends a a result of w muller and s marshall who proved the corresponding statement for closed arithmetic 3 manifolds to the finite volume case we also prove a limit multiplicity formula for combinatorial reidemeister torsions on higher dimensional hyperbolic manifolds

robust maximum principle for multi model lq problem this paper presents the version of the robust maximum principle in the context of multi model control formulated as the minimax bolza problem the cost function contains a terminal term as well as an integral one a fixed horizon and terminal set are considered the necessary conditions of the optimality are derived for the class of uncertain systems given by an ordinary differential equation with parameters from a given finite set this problem consists in the control design providing a good behaviour for a given class of multi model system it is shown that the design of the minimax optimal controller is reduced to a finite dimensional optimization problem given at the corresponding simplex set containing the weight parameters to be found the robust optimal control may be interpreted as a mixture with the optimal weights of the controls which are optimal for each fixed parameter value the proof is based on the recent results obtained for minimax mayer problem boltyanski and poznyak 1999a the mini

generalisation of euler lagrange equations to find min max optimal solution of uncertain systems a major problem in optimal control theory is existing uncertainty in initial state cost function or in state equations in this paper a generalised euler lagrange approach to find min max optimal solution of uncertain systems with uncertain or certain cost function using calculus of variation is considered the work is based on an admissible system path or a trajectory which minimises the maximum value of the functional over all uncertainty first of all a new form of euler lagrange conditions for uncertain systems is presented then several cases are indicated where final condition can be specified or free also necessary conditions are introduced to existence of min max optimal solution of the uncertain systems also the method is generalised when the uncertainties are bounded with using pontryagin s minimum principle finally efficiency of the proposed methods is verified through some examples

selberg zeta functions on odd dimensional hyperbolic manifolds of finite volume we study selberg zeta functions z s associated to locally homogeneous vector bundles over the unit sphere bundle of a complete odd dimensional hyperbolic manifold of finite volume we assume a certain condition on the fundamental group of the manifold a priori the selberg zeta functions are defined only for s in some right halfspace of c we will prove that for any locally homogeneous bundle the functions z s have a meromorphic continuation to c and we will give a complete description of their singularities in terms of spectral data of the underlying manifold our work generalizes results of bunke and olbrich to the non compact situation as an application of our results one can compare the normalized reidemeister torsions on hyperbolic 3 manifolds with cusps which were introduced by menal ferrer and porti to the corresponding regularized analytic torsions

robust optimal feedback for terminal linear quadratic control problems under disturbances we consider a linear dynamic system in the presence of an unknown but bounded perturbation and study how to control the system in order to get into a prescribed neighborhood of a zero at a given final moment the quality of a control is estimated by the quadratic functional we define optimal guaranteed program controls as controls that are allowed to be corrected at one intermediate time moment we show that an infinite dimensional problem of constructing such controls is equivalent to a special bilevel problem of mathematical programming which can be solved explicitely an easy implementable algorithm for solving the bilevel optimization problem is derived based on this algorithm we propose an algorithm of constructing a guaranteed feedback control with one correction moment we describe the rules of computing feedback which can be implemented in real time mode the results of illustrative tests are given

min max control of constrained uncertain discrete time linear systems for discrete time uncertain linear systems with constraints on inputs and states we develop an approach to determine state feedback controllers based on a min max control formulation robustness is achieved against additive norm bounded input disturbances and or polyhedral parametric uncertainties in the state space matrices we show that the finite horizon robust optimal control law is a continuous piecewise affine function of the state vector and can be calculated by solving a sequence of multiparametric linear programs when the optimal control law is implemented in a receding horizon scheme only a piecewise affine function needs to be evaluated on line at each time step the technique computes the robust optimal feedback controller for a rather general class of systems with modest computational effort without needing to resort to gridding of the state space

robust maximum principle in minimax control this paper presents a new version of the maximum principle dealing with construction of optimal control strategies for the class of uncertain systems given by an ordinary differential equation with unknown parameters from a given finite set the problem under consideration belongs to the class of optimization problems of the minimax type the proof is based on the tent method illustrative examples conclude this study

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

exponential growth of torsion in the cohomology of arithmetic hyperbolic manifolds for d 2n 1 a positive odd integer we consider sequences of arithmetic subgroups of text so 0 d 1 and yielding corresponding hyperbolic manifolds of finite volume and show that under appropriate and natural assumptions the torsion of the associated cohomology groups grows exponentially

analytic torsion and r torsion for unimodular representations let m be a closed c manifold of dimension n both r torsion and analytic torsion are smooth invariants of acyclic orthogonal or unitary representations p of the fundamental group 7i m the reidemeister franz torsion or r torsion tm p of p is defined in terms of the combinatorial structure of m given by its smooth triangulations the analytic torsion tm p was introduced by ray and singer rs as an analytic counterpart of r torsion in order to define the analytic torsion one has to choose a riemannian metric on m then tm p is a certain weighted alternating product of regularized determinants of the laplacians on differential qforms of m with values in the flat bundle e defined by p it was conjectured by ray and singer rs that tm p trm p for all acyclic orthogonal or unitary representations p this conjecture was proved independently by cheeger c and the author mu the restriction to orthogonal or unitary representations is certainly a limitation of the applicability of this result if 7i m is infinite because an infinite discrete group will have in general many nonorthogonal finite dimensional representations it is the purpose of the present paper to remove this limitation we call a representation p r m gl e on a finite dimensional real or complex vector space e unimodular if i det p y i 1 for all y e i m then we define r torsion and analytic torsion for unimodular representations and the main result is that for odd dimensional manifolds m the equality of the two torsions extends to all unimodular representations

the analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds of finite volume abstract in this paper we study the regularized analytic torsion of finite volume hyperbolic manifolds we consider sequences of coverings x i of a fixed hyperbolic orbifold x 0 our main result is that for certain sequences of coverings and strongly acyclic flat bundles the analytic torsion divided by the index of the covering converges to the l 2 torsion our results apply to certain sequences of arithmetic groups in particular to sequences of principal congruence subgroups of so 0 d 1 z and to sequences of principal congruence subgroups or hecke subgroups of bianchi groups

on the worst case disturbance of minimax optimal control we consider a minimax optimal control problem for uncertain stochastic systems the uncertainty in the underlying stochastic system is formulated in terms of probability measure perturbations satisfying a relative entropy constraint by characterizing the worst case measure for a related stochastic minimax game it is shown that the worst case uncertain system can be represented in the form of a parametric perturbation of the nominal system a numerical example is presented to illustrate theoretical results developed in this paper

min max feedback model predictive control for constrained linear systems min max feedback formulations of model predictive control are discussed both in the fixed and variable horizon contexts the control schemes the authors discuss introduce in the control optimization the notion that feedback is present in the receding horizon implementation of the control this leads to improved performance compared to standard model predictive control and resolves the feasibility difficulties that arise with the min max techniques that are documented in the literature the stabilizing properties of the methods are discussed as well as some practical implementation details

a numerical solution of open loop nash equilibrium in nonlinear differential games based on chebyshev pseudospectral method in general the applications of differential games for solving practical problems have been limited because all calculations had to be done analytically in this investigation a simple and efficient numerical method for solving nonlinear nonzero sum differential games with finite and infinite time horizon is presented in both cases derivation of open loop nash equilibria solutions usually leads to solving nonlinear boundary value problems for a system of odes the proposed numerical method is based on a combination of minimum principle of pontryagin and expanding the required approximate solutions as the elements of chebyshev polynomials applying chebyshev pseudospectral method two point boundary value problems in differential games are reduced to the solution of a system of algebraic equations finally several examples are given to demonstrate the accuracy and efficiency of the proposed method and a comparison is made with the results obtained by fourth order runge kutta method

min max sliding mode control for multimodel linear time varying systems an original linear time varying system with unmatched disturbances and uncertainties is replaced by a finite set of dynamic models such that each one describes a particular uncertain case including exact realizations of possible dynamic equations as well as external bounded disturbances such a tradeoff between an original uncertain linear time varying dynamic system and a corresponding higher order multimodel system with a complete knowledge leads to a linear multi model system with known bounded disturbances each model from a given finite set is characterized by a quadratic performance index the developed min max sliding mode control strategy gives an optimal robust sliding surface design algorithm which is reduced to a solution of an equivalent linear quadratic problem that corresponds to the weighted performance indices with weights from a finite dimensional simplex an illustrative numerical example is presented

design of receding horizon controls for constrained time varying systems in this note we propose a generalized stabilizing receding horizon control scheme for input state constrained linear discrete time varying systems that improves feasibility and on line computation on the constrained finite horizon optimization problem compared with existing schemes the control scheme is based on a time varying horizon cost function with time varying terminal weighting matrices which can easily be implemented via linear matrix inequality technique we discuss modifications of the proposed scheme that improve feasibility or on line computation time through simulation examples we illustrate the results of these schemes

suboptimal model predictive control feasibility implies stability practical difficulties involved in implementing stabilizing model predictive control laws for nonlinear systems are well known stabilizing formulations of the method normally rely on the assumption that global and exact solutions of nonconvex nonlinear optimization problems are possible in limited computational time in the paper we first establish conditions under which suboptimal model predictive control mpc controllers are stabilizing the conditions are mild holding out the hope that many existing controllers remain stabilizing even if optimality is lost second we present and analyze two suboptimal mpc schemes that are guaranteed to be stabilizing provided an initial feasible solution is available and for which the computational requirements are more reasonable

min max optimal control of linear systems with uncertainty and terminal state constraints in this paper a class of min max optimal control problems with continuous dynamical systems and quadratic terminal constraints is studied the main contribution is that the original terminal state constraint in which the disturbance is involved is transformed into an equivalent linear matrix inequality without disturbance under certain conditions then the original min max optimal control problem is solved via solving a sequence of semi definite programming problems an example is presented to illustrate the proposed method

control of constrained discrete time systems with bounded ell_ 2 gain we consider the problem of designing a control law for a constrained linear system with bounded disturbances that ensures constraint satisfaction over an infinite horizon while also guaranteeing that the closed loop system has bounded lscr2 gain to this end we propose a receding horizon control strategy based on the repeated calculation of optimal finite horizon feedback policies we parameterize these policies such that the input at each time is an affine function of current and prior states and minimize a worst case quadratic cost where the disturbance energy is negatively weighted as in h infin control we show that the resulting receding horizon controller has two advantages over previous results for this problem first the policy optimization problem to be solved at each time step can be rendered convex concave with a number of decision variables and constraints that grows polynomially with the problem size thereby making its solution amenable to standard techniques in convex optimization second the achievable lscr2 gain of the resulting closed loop system is bounded and non increasing with increasing control horizon a numerical example is included to demonstrate the improvement in achievable lscr2 gain relative to existing methods

an iterative adaptive dynamic programming method for solving a class of nonlinear zero sum differential games in this paper a new iterative adaptive dynamic programming adp method is proposed to solve a class of continuous time nonlinear two person zero sum differential games the idea is to use the adp technique to obtain the optimal control pair iteratively which makes the performance index function reach the saddle point of the zero sum differential games if the saddle point does not exist the mixed optimal control pair is obtained to make the performance index function reach the mixed optimum stability analysis of the nonlinear systems is presented and the convergence property of the performance index function is also proved two simulation examples are given to illustrate the performance of the proposed method

the explicit constrained min max model predictive control of a discrete time linear system with uncertain disturbances in this technical brief we develop an algorithm to determine the explicit solution of the constrained min max model predictive control problem for a discrete time linear system with bounded additive uncertain disturbance the control law is determined to be piecewise affine from a quadratic cost function and the state space is partitioned into corresponding polyhedral cones by moving the on line implementation to an off line explicit evaluation the computational burden is decreased and the applicability of min max optimization is broadened the results of this approach are shown via computer simulations