convergence results for smooth regularizations of hybrid nonlinear optimal control problems we consider a class of hybrid nonlinear optimal control problems having a discontinuous dynamics ruled by a partition of the state space for this class of problems some hybrid versions of the usual pontryagin maximum principle are known we introduce general regularization procedures parameterized by a small parameter smoothing the previous hybrid problems to standard smooth optimal control problems for which we can apply the usual pontryagin maximum principle we investigate the question of the convergence of the resulting extremals as the regularization parameter tends to zero under some general assumptions we prove that smoothing regularization procedures converge in the sense that the solution of the regularized problem as well as its extremal lift converges to the solution of the initial hybrid problem to illustrate our convergence result we apply our approach to the minimal time low thrust coplanar orbit transfer with eclipse constraint

nonconvex duality in optimal control we derive a representation of the minimum cost of an optimal control problem in terms of the upper envelope of generalized semisolutions of the hamilton jacobi equation

deterministic near optimal controls part ii dynamic programming and viscosity solution approach near optimization is as sensible and important as optimization for both theory and applications this paper concerns dynamic near optimization or near optimal controls for systems governed by deterministic ordinary differential equations and uses dynamic programming to study the near optimality since nonsmoothness is inherent in this subject the viscosity solution approach is employed to investigate the problem the dynamic programming equation is derived in terms of e superdifferential subdifferential the relationships among the adjoint functions the value functions and the hamiltonian along near optimal trajectories are revealed verification theorems with which near optimal feedback controls can be constructed are obtained

on the synthesis of a stabilizing feedback control via lie algebraic methods let the n dimensional system of differential equations dx dt x x t have p in mathbb r n as a rest solution i e x p 0 even in cases when this rest solution is unstable one can often induce a strong stability asymptotic stability by the inclusion of one or more controls e g via a controlled system a dx dt x x uy x where say u leqq 1 lie theory gives a computable sufficient condition to determine when by use of the control u one can steer a full n dimensional neighborhood of p to p by solutions of a this condition is assumed to hold one prefers a feedback control i e that u u x the main result in this paper is an algorithm which determines a modified stabilizing feedback control specifically for given varepsilon 0 one measures the current state q and the algorithm determines u t q 0 leqq t leqq varepsilon such that the solution x cdot u of a initiating from q and corresponding to this control u satisfies distance x varepsilon u p q p in fact iterates are theoretically shown to converge to p numerical examples computed via a simple fortran program are included these substantiate the strong stability achieved via such a modified feedback control

a general theorem on local controllability we prove a general sufficient condition for local controllability of a nonlinear system at an equilibrium point earlier results of brunovsky hermes jurdjevic crouch and byrnes sussmann and grossmann are shown to be particular cases of this result also a number of new sufficient conditions are obtained all these results follow from one simple general principle namely that local controllability follows whenever brackets with certain symmetries can be neutralized in a suitable way by writing them as linear combinations of brackets of a lower degree both the class of symmetries and the definition of degree can be chosen to suit the problem

smooth regularization of bang bang optimal control problems consider the minimal time control problem for a single input control affine system x x x u1y1 x in rn where the scalar control u1 satisfies the constraint lu1 l when applying a shooting method for solving this kind of optimal control problem one may encounter numerical problems due to the fact that the shooting function is not smooth whenever the control is bang bang in this article we propose the following smoothing procedure for e 0 small we consider the minimal time problem for the control system x x x u1ey1 x e i 2m uieyi x where the scalar controls uie i 1 m with m 2 satisfy the constraint i 1m uie uie t 2 1 we prove under appropriate assumptions a strong convergence result of the solution of the regularized problem to the solution of the initial problem

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

brief exponential stabilization of nonholonomic dynamic systems by smooth time varying control a general dynamic model is proposed for describing a large class of nonholonomic systems including extended chained systems extended power systems underactuated surface vessel systems etc by introducing an assistant state variable and a time varying state transformation based on the concept of minimal dilation degree this class of nonholonomic systems is transformed into linear time varying control systems and the asymptotic exponential stability is thus achieved by using a smooth time varying feedback control law the existence and uniqueness of the minimal dilation degree for the discussed systems are also proved under certain conditions

effect of nonlinear correlations on the statistics of return intervals in multifractal data sets we study the statistics of return intervals between events above a certain threshold in multifractal data sets without linear correlations we find that nonlinear correlations in the record lead to a power law i decay of the autocorrelation function of the return intervals ii increase in the conditional return period and iii decay in the probability density function of the return intervals we show explicitly that all the observed quantities depend both on the threshold value and system size and hence there is no simple scaling observed we also demonstrate that this type of behavior can be observed in real economic records and can be used to improve considerably risk estimation

the dual arc inclusion with differential inclusions this paper continues our study of the minimum time problem with dynamics driven by a particular type of differential inclusion we prove under certain assumptions that the dual arc associated to an optimal solution and the supergradient of the value function along the solution are intertwined in this new setting we also derive new optimality conditions that contain nonsmooth features with an additional constancy condition on the hamiltonian

local stabilization of minimum phase nonlinear systems abstract in this note we briefly review the notion of a nonlinear minimum phase system i e a system which when constrained in such a way as to produce zero output evolves with an asymptotically stable dynamics the main purpose of the paper is to show that any minimum phase nonlinear system can always be locally stabilized by smooth state feedback

diffusion of epicenters of earthquake aftershocks omori s law and generalized continuous time random walk models the epidemic type aftershock sequence etas model is a simple stochastic process modeling seismicity based on the two best established empirical laws the omori law power law decay ensuremath sim 1 t 1 ensuremath theta of seismicity after an earthquake and gutenberg richter law power law distribution of earthquake energies in order to describe also the space distribution of seismicity we use in addition a power law distribution ensuremath sim 1 r 1 ensuremath mu of distances between triggered and triggering earthquakes the etas model has been studied for the last two decades to model real seismicity catalogs and to obtain short term probabilistic forecasts here we present a mapping between the etas model and a class of ctrw continuous time random walk models based on the identification of their corresponding master equations this mapping allows us to use the wealth of results previously obtained on anomalous diffusion of ctrw after translating into the relevant variable for the etas model we provide a classification of the different regimes of diffusion of seismic activity triggered by a mainshock specifically we derive the relation between the average distance between aftershocks and the mainshock as a function of the time from the mainshock and of the joint probability distribution of the times and locations of the aftershocks the different regimes are fully characterized by the two exponents ensuremath theta and ensuremath mu our predictions are checked by careful numerical simulations we stress the distinction between the bare omori law describing the seismic rate activated directly by a mainshock and the renormalized omori law taking into account all possible cascades from mainshocks to aftershocks of aftershock of aftershock and so on in particular we predict that seismic diffusion or subdiffusion occurs and should be observable only when the observed omori exponent is less than 1 because this signals the operation of the renormalization of the bare omori law also at the origin of seismic diffusion in the etas model we present predictions and insights provided by the etas to ctrw mapping which suggest different ways for studying seismic catalogs finally we discuss the present evidence for our predicted subdiffusion of seismicity triggered by a main shock stressing the caveats and limitations of previous empirical works

maple code for the gamma algorithm for global optimization of uncertain functions in economy and finance problems with uncertainties are ubiquitous in many areas of life and the economy due to a lack of information as regards the economy and in finance problems with uncertainties stock prices marketing problems inflation unemployment are usually formulated by giving bounds on maximum and minimum values of certain parameters i e box constraints in such situations it is necessary to make a choice of better parameters that produce finite intervals of possible values for a given uncertain function at each point of the parameter space the gamma algorithm presents a method for making that choice a variant of the gamma algorithm based on the cubic algorithm is considered for global optimization of uncertain functions with box constraints in r n the set monotonic algorithm contains a block for problems with equality constraints and operates within the unit cube 0 1 n for all problems on this basis a maple code of modular structure is developed for full global optimization of uncertain functions in n variables the code does not create ill conditioned situations graphics are included and the solution set can be visualized in plane projections and sections an example related to minsky s financial instability hypothesis is presented with a graph to illustrate the use of the code

dynamic programming principle of control systems on manifolds and its relations to maximum principle abstract we study the dynamic programming principle dpp for short on manifolds obtain the hamilton jacobi bellman hjb for short equation and prove that the value function is the only viscosity solution to the hjb equation then we investigate the relation between dpp and pontryagin s maximum principle pmp for short from which we obtain pmp on manifolds

local equilibrium controllability of multibody systems controlled via shape change we study local equilibrium controllability of shape controlled multibody systems the multibody systems are defined on a trivial principal fiber bundle by a lagrangian that characterizes the base body motion and shape dynamics a potential dependent on an advected parameter e g uniform gravitational potential is considered this potential breaks base body symmetries but a symmetry subgroup is assumed to exist symmetric product formulas are derived and important properties are obtained for symmetric products of horizontal shape control vector fields and a potential vector field that is dependent on an advected parameter based on these properties sufficient conditions for local equilibrium controllability and local fiber equilibrium controllability are developed these results are applied to two classes of shape controlled multibody systems in a uniform gravitational field multibody attitude systems and neutrally buoyant multibody underwater vehicles

adaptive output feedback stabilization for nonholonomic systems with strong nonlinear drifts abstract this paper investigates the design of an output feedback adaptive stabilization controller for a nonholonomic chained system with strong nonlinear drifts including modeled nonlinear dynamics unmodeled dynamics and dynamics modeled with unknown parameters also the virtual control directions of the system are unknown the purpose is to design a nonlinear output feedback switching controller such that the closed loop system is globally asymptotically stable a novel observer and estimator are introduced for states and parameter estimates respectively a constructive procedure of design for an output feedback adaptive controller is given by using the integrator backstepping approach and based on the proposed observer and parameter estimator an example is given to show the effectiveness of the proposed scheme

the isaacs equation for differential games totally optimal fields of trajectories and related problems some aspects of the isaacs principle of transition and his main equation for differential games are considered to demonstrate that those equations contain some implicit assumptions and are valid under certain contiguity condition which is defined and analyzed for differential games the notion of total optimality is defined and the totally optimal fields of trajectories and control curves are introduced and studied in relation to the isaacs principle of transition bellman principle of optimality maximum principle of pontryagin and variational principles of mechanics it is demonstrated that the isaacs bellman and pontryagin theories are valid if and only if the optimal trajectories and optimal control curves generated by those methods are totally optimal in this context the hamilton jacobi partial differential equation can be used for sequential solution of multi games defined as n person games with m controls r cost functionals and multiple min max min max etc operators in fixed order of application and not creating multi objective game problems over totally optimal fields the structure of controls is invariant under time uncertainty parallel and series games are considered so the isaacs procedure can be reduced to the application of the bellman equations twice control systems with incomplete information or structural limitations on controls do not in general satisfy the contiguity condition thus are not totally optimal game problems for such systems may have optimal solutions which however cannot be obtained by the isaacs equations this fact is shown in an example of a widely used engineering system for which an optimal trajectory has all its parts non optimal and non contiguous to the optimal trajectory the paper presents theoretical justification of the isaacs equations for contiguous systems comparison of optimal control principles with variational principles of mechanics the consideration of total optimality and totally optimal fields of trajectories as necessary and sufficient conditions for validity of the three major optimal control theories and some other results important for applications

a partial differential inequality for dissipative nonlinear systems abstract in this paper we show that storage functions for dissipative systems can be characterised by a partial differential inequality in the weak or viscosity sense the storage functions need not be differentiable this characterisation is stable with respect to perturbations we formulate a version of the positive real lemma for passive systems using the weak sense framework

exponential stabilization of an underactuated autonomous surface vessel this paper studies the problem of controlling the planar position and orientation of an autonomous surface vessel using two independent thrusters it is first shown that although the system is not asymptotically stabilizable to a given configuration using a time invariant continuous feedback it is strongly accessible and small time locally controllable at any equilibrium time invariant discontinuous feedback laws are then constructed to asymptotically stabilize the system to the desired configuration with exponential convergence rates a simulation example is included to demonstrate the results

passivity and feedback passification of switched discrete time linear systems abstract this paper focuses on the passivity analysis and feedback passification for switched discrete time linear systems using multiple storage functions under a certain switching law controllers and switching laws are designed based on complete and partial state measurements respectively conditions for strict passivity of a switched discrete time system are obtained without the requirement of strict passivity of subsystems in the case of partial state measurements dynamic output feedback controllers for subsystems are designed and a switching law depending only on the state of the output feedback controllers is constructed to guarantee strict passivity of the closed loop switched system finally a numerical example is provided to demonstrate the feasibility of the theoretical results

input output to state stability this work explores lyapunov characterizations of the input output to state stability ioss property for nonlinear systems the notion of ioss is a natural generalization of the standard zero detectability property used in the linear case the main contribution of this work is to establish a complete equivalence between the ioss property and the existence of a certain type of smooth lyapunov function as corollaries one shows the existence of norm estimators and obtains characterizations of nonlinear detectability in terms of relative stability and of finite energy estimates

passivity and stability of switched systems a multiple storage function method this paper presents a concept of passivity for switched systems using multiple storage functions this passivity property is invariant under compatible feedback interconnection branicky s stability theorem of multiple lyapunov functions is generalized by relaxing the non increasing condition on values of lyapunov like functions using this result we show that a passive switched system is stable in the sense of lyapunov moreover asymptotic stability is reached if all subsystems are asymptotically detectable

steering for a class of dynamic nonholonomic systems we derive control algorithms for a class of dynamic nonholonomic steering problems characterized as mechanical systems with nonholonomic constraints and symmetries we use a simplified geometric framework in developing two types of control algorithms for such systems the first targets local configuration controllability using a perturbation approach to establish results similar to steering using sinusoids the second method builds on this approach by utilizing kinematic steering algorithms for mobile robots in generating more nonlocal motion plans for this class of systems

value function relaxation and transversality conditions in infinite horizon optimal control abstract we investigate the value function v r r n r of the infinite horizon problem in optimal control for a general not necessarily discounted running cost and provide sufficient conditions for its lower semicontinuity continuity and local lipschitz regularity then we use the continuity of v t to prove a relaxation theorem and to write the first order necessary optimality conditions in the form of a possibly abnormal maximum principle whose transversality condition uses limiting horizontal supergradients of v 0 at the initial point when v 0 is merely lower semicontinuous then for a dense subset of initial conditions we obtain a normal maximum principle augmented by sensitivity relations involving the frechet subdifferentials of v t finally when v is locally lipschitz we prove a normal maximum principle together with sensitivity relations involving generalized gradients of v for arbitrary initial conditions such relations simplify drastically the investigation of the limiting behavior at infinity of the adjoint state

dissipativity theory for switched systems a framework of dissipativity theory for switched systems using multiple storage functions and multiple supply rates is set up each subsystem of a switched system is associated with a storage function to describe the energy stored in the subsystem and is associated with a supply rate that represents energy coming from outside the subsystem when the subsystem is active the exchange of energy between the active subsystem and an inactive subsystem is characterized by cross supply rates stability is reached when all supply rates can be made negative as long as the total exchanged energy between the active subsystem and any inactive subsystems is finite in some sense two special forms of dissipativity passivity and l 2 gain are addressed for both cases asymptotic stability is guaranteed under certain negative output feedback plus asymptotic zero state detectability switched passivity conditions and switched l 2 gain inequalities are respectively derived which are generalizations of classical ones feedback invariance of passivity and a small gain theorem are also given

on trajectory and force tracking control of constrained mobile manipulators with parameter uncertainty this paper studies the trajectory and force tracking control problem of mobile manipulators subject to holonomic and nonholonomic constraints with unknown inertia parameters adaptive controllers are proposed based on a suitable reduced dynamic model the defined reference signals and the mixed tracking errors the proposed controllers not only ensure the entire state of the system to asymptotically converge to the desired trajectory but also ensure the constraint force to asymptotically converge to the desired force a detailed numerical example is presented to illustrate the developed methods

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

bifurcation and chaotic behavior of a discrete singular biological economic system abstract in this paper the complex dynamics of a discrete singular biological economic system are first investigated firstly we establish a discrete singular biological economic system which is based on the discretization of a differential algebraic equations that is described by a ratio dependent predator prey system with harvesting and economic factor then it is shown that the system undergoes flip bifurcation and neimark sacker bifurcation in the interior of r 3 by using the new normal form of discrete singular systems the center manifold theorem and the bifurcation theory as varying the economic profit in some range numerical simulations are presented not only to illustrate our results with the theoretical analysis but also to exhibit the complex dynamic behaviors such as cascades of period doubling bifurcation in orbits of period 2 4 8 and chaotic sets these results reveal far richer dynamics of the discrete model compared with the continuous model the lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors

memory in the occurrence of earthquakes we study the statistics of the recurrence times ensuremath tau between earthquakes above a certain magnitude m in six one global and five regional earthquake catalogs we find that the distribution of the recurrence times strongly depends on the previous recurrence time ensuremath tau 0 such that small and large recurrence times tend to cluster in time this dependence on the past is reflected in both the conditional mean recurrence time and the conditional mean residual time until the next earthquake which increase monotonically with ensuremath tau 0 as a consequence the risk of encountering the next event within a certain time span after the last event depends significantly on the past an effect that has to be taken into account in any effective earthquake prognosis

chaperoning state evolutions by variable durations mckendrick partial differential equations in demography provide the state of a population depending on chronological time t and age d and involve two time evolutions the chronological one t in 0 t mapsto t geq 0 and the calendar age t in t d t mapsto t t d geq 0 both with constant velocity equal to 1 the calendar age evolution chaperons so to speak the evolution t in t d t mapsto x t of the state of a system some physical biological and economic problems motivate the introduction of variable durations with variable velocities representing the fluidity of time offering mathematical metaphors of a subjective fleeting specious time passing more or less slowly variable durations are no longer prescribed but chosen among those available and regulated the joint evolution of the variable evolution and the state is assumed to be governed by a differential inclusion or control system and provides as a byproduct the unknown temporal windows on which they evolve toge

a connection between the maximum principle and dynamic programming for constrained control problems we consider the mayer optimal control problem with dynamics given by a nonconvex differential inclusion whose trajectories are constrained to a closed set and obtain necessary optimality conditions in the form of the maximum principle together with a relation between the costate and the value function this additional relation is applied in turn to show that the maximum principle is nondegenerate we also provide a sufficient condition for the normality of the maximum break principle to derive these results we use convex linearizations of differential inclusions and convex linearizations of constraints along optimal trajectories then duality theory of convex analysis is applied to derive necessary conditions for optimality in this way we extend the known relations between the maximum principle and dynamic programming from the unconstrained problems to the constrained case

asymptotic approach on conjugate points for minimal time bang bang controls abstract we focus on the minimal time control problem for single input control affine systems x x x u 1 y 1 x in r n with fixed initial and final time conditions x 0 x 0 x t f x 1 and where the scalar control u 1 satisfies the constraint u 1 1 for these systems a concept of conjugate time t c has been defined in e g agrachev et al 2002 23 maurer and osmolovskii 2004 21 and noble and schattler 2002 28 in the bang bang case besides theoretical and practical issues for conjugate time theory are well known in the smooth case see e g agrachev and sachkov 2004 43 and milyutin and osmolovskii 1998 15 and efficient implementation tools are available see bonnard et al 2007 35 the first conjugate time along an extremal is the time at which the extremal loses its local optimality in this work we use the asymptotic approach developed in silva and trelat in press 36 and investigate the convergence properties of conjugate times more precisely for e 0 small and arbitrary vector fields y 1 y m we consider the minimal time problem for the control system x e x x e u 1 e y 1 x e e i 2 m u i e y i x e under the constraint i 1 m u i e 2 1 with the fixed boundary conditions x e 0 x 0 x e t f x 1 of the initial problem under appropriate assumptions the optimal controls of the latter regularized optimal control problem are smooth and the computation of associated conjugate times t c e falls into the standard theory our main result asserts the convergence as e tends to 0 of t c e towards the conjugate time t c of the initial bang bang optimal control problem as well as the convergence of the associated extremals as a byproduct we obtain an efficient algorithmic way to compute conjugate times in the bang bang case

a patchy dynamic programming scheme for a class of hamilton jacobi bellman equations in this paper we present a new algorithm for the solution of hamilton jacobi bellman equations related to optimal control problems the key idea is to divide the domain of computation into subdomains which are shaped by the optimal dynamics of the underlying control problem this can result in a rather complex geometrical subdivision but it has the advantage that every subdomain is invariant with respect to the optimal dynamics and then the solution can be computed independently in each subdomain the features of this dynamics dependent domain decomposition can be exploited to speed up the computation and for an efficient parallelization since the classical transmission conditions at the boundaries of the subdomains can be avoided for their properties the subdomains are patches in the sense introduced by ancona and bressan esaim control optim calc var 4 1999 pp 445 471 several examples in two and three dimensions illustrate the properties of the new method

algorithm for overcoming the curse of dimensionality for state dependent hamilton jacobi equations abstract in this paper we develop algorithms to overcome the curse of dimensionality in non convex state dependent hamilton jacobi partial differential equations hj pdes arising from optimal control and differential game problems the subproblems are independent and they can be implemented in an embarrassingly parallel fashion this is ideal for perfect scaling in parallel computing the algorithm is proposed to overcome the curse of dimensionality 1 2 when solving hj pde the major contribution of the paper is to change either the solving of a pde problem or an optimization problem over a space of curves to an optimization problem of a single vector which goes beyond the work of 40 we extend the method in 7 9 15 and conjecture a lax type minimization principle to solve state dependent hj pde when the hamiltonian is convex as well as a hopf type maximization principle to solve state dependent hj pde when the hamiltonian is non convex as a generalization of the well known hopf formula in 18 25 50 we showed the validity of the formula under restricted assumption for the sake of completeness and would like to bring our readers to 62 which validates our conjectures in a more general setting we conjectured the weakest assumption of our formula to hold is a pseudoconvexity assumption similar to one stated in 50 our method is expected to have application in control theory differential game problems and elsewhere

the cubic algorithm for global games with application to pursuit evasion games abstract new algorithms for solution of nonconvex global games defined over a cube are presented application to differential games and to pursuit evasion games is considered and extension is made for games over general compact and robust sets defined by constraints that may depend on time and on state variables the methods are deterministic and monotonically set convergent to deliver in the limit the unique exact full globally optimal minimax and maximin solutions for both players or the entire global saddle set if it exists a stopping rule is provided for determining approximate solutions with a given precision in a finite number of iterations a modification is considered to play on mistakes of the opponent on line replacement of cost functionals is possible in the course of iterations according to changing interests of the players no separability nor convexity concavity assumptions are imposed on the cost functional and variational methods are not used the ideas are illustrated by examples and application is made to determining the globally optimal closed loop strategies for the ship torpedo collision avoidance differential game with manoeuvrability constraints

stabilizability of nonholonomic control systems abstract we consider here the problem of stabilization of nonlinear hamiltonian control systems with nonholonomic constraints such systems are of interest in various applications including the control of robotic systems we discuss the formulation of hamiltonian systems with nonholonomic constraints in particular considering the differences between the dirac theory of constraints the classical theory of nonholonomic constraints and weber s theory of generalized nonholonomic constraints stabilization of systems with these constraints is discussed using the theory of hamiltonian control systems and a reduced stabilization criterion is given

isoholonomic problems and some applications we study the problem of finding the shortest loops with a given holonomy we show that the solutions are the trajectories of particles in yang mills potentials theorem 4 or equivalently the projections of kaluza klein geodesics theorem 2 applications to quantum mechanics berry s phase sect 3 and the optimal control of deformable bodies sect 6 are touched upon

unified scaling law in the coherent noise model abstract the waiting time distribution between successive events and the unified scaling law is studied using the coherent noise model it is shown that although this model generates uncorrelated event sizes and does not exhibit criticality it still provides the unified scaling law we argue the role of characteristic kink observed in the unified scaling law and the meaning of the parameter c used to fix the peak of the kink to unity our results indicate that the parameter c is indeed a physical quantity localizing the end of the linear tendency in the scaling law which corresponds to the completion of the dominance of correlated events in time

the turnpike property in finite dimensional nonlinear optimal control turnpike properties have been established long time ago in finite dimensional optimal control problems arising in econometry they refer to the fact that under quite general assumptions the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces the first and the last of which being transient short time arcs and the middle piece being a long time arc staying exponentially close to the optimal steady state solution of an associated static optimal control problem we provide in this paper a general version of a turnpike theorem valuable for nonlinear dynamics without any specific assumption and for very general terminal conditions not only the optimal trajectory is shown to remain exponentially close to a steady state but also the corresponding adjoint vector of the pontryagin maximum principle the exponential closedness is quantified with the use of appropriate normal forms of riccati equations we show then how the property on the adjoint vector can be adequately used in order to initialize successfully a numerical direct method or a shooting method in particular we provide an appropriate variant of the usual shooting method in which we initialize the adjoint vector not at the initial time but at the middle of the trajectory

some old traditions in mathematics and in mathematical education abstract continuing the discussion by azbelev about the harmful influence of some classical traditions on the study of differential equations 1 this note presents further instances of old traditions and of their impact on the development of certain areas of mathematics and its applications

a note on the value function for constrained control problems abstract we consider the mayer optimal control problem with dynamics given by a nonconvex differential inclusion whose trajectories are constrained to a given set and we obtain a relation between the costate function that appears in the maximum principle and the value function this relation extends the known conditions existing in the literature for unconstrained problems to those for problems under state constraints

combining pontryagin s principle and dynamic programming for linear and nonlinear systems to study optimal control and disturbance attenuation problems two prominent and somewhat alternative strategies have emerged in the last century dynamic programming dp and pontryagin s minimum principle pmp the former characterizes the solution by shaping the dynamics in a closed loop a priori unknown via the selection of a feedback input at the price however of the solution to typically daunting partial differential equations the latter instead provides extended dynamics that must be satisfied by the optimal process for which boundary conditions a priori unknown should be determined the results discussed in this article combine the two approaches by matching the corresponding trajectories i e combining the underlying sources of information knowledge of the complete initial condition from dp and of the optimal dynamics from pmp the proposed approach provides insights for linear as well as nonlinear systems in the case of linear systems the derived conditions lead to matrix algebraic equations similar to the classic algebraic riccati equations ares although with coefficients defined as polynomial functions of the input gain matrix with the property that the coefficient of the quadratic term of such equation is sign definite even if the corresponding coefficient of the original are is sign indefinite as it is typically the case in the mathcal h infty control problem this feature is particularly appealing from the computational point of view since it permits the use of standard minimization techniques for convex functions such as the gradient algorithm in the presence of nonlinear dynamics the strategy leads to algebraic equations that allow us to locally construct the optimal feedback by considering the behavior of the closed loop dynamics at a single point in the state space

output feedback adaptive stabilization control design for non holonomic systems with strong non linear drifts this paper investigates the problem of output feedback adaptive stabilization control design for non holonomic chained systems with strong non linear drifts including modelled non linear dynamics unmodelled dynamics and those modelled but with unknown parameters an observer and an estimator are introduced for state and parameter estimates respectively by using the integrator backstepping approach and based on the observer and parameter estimator a constructive design procedure for output feedback adaptive stabilization control is given it is shown that under some conditions the control design ensures the closed loop system is globally asymptotically stable when there is no non linear drift in the first subsystem and semiglobally asymptotically stable otherwise an example is given to show the effectiveness of the theory

the beta algorithm abstract a set monotonic non gradient algorithm is proposed for finding global minima of general non convex mathematical programming problems the algorithm is based on the cubic algorithm e a galperin j math anal appl m 112 1985 635 640 equipped with a semi certain distinction operator and the marginal comparison constant generator

non convex global optimization by the beta algorithm a maple code abstract a variant of the beta algorithm based on cubic algorithm math comput modelling 38 2003 77 97 is presented for global optimization of continuous functions over general compact robust sets this is a set monotonic algorithm over non convex disconnected set not satisfying any qualification constraint other than being compact and robust on this basis a maple code is developed for full global optimization of functions of n variables the code does not create ill conditioned situations graphics are included and the solution set can be visualized in projections on coordinate planes the code is ready for engineering applications results of numerical experiments are presented with graphs to illustrate the use of the code

self similar branching of aftershock sequences abstract in this paper we propose a branching aftershock sequence bass model for seismicity we suggest that the bass model is a preferred alternative to the widely studied epidemic type aftershock sequence etas model in the bass model an initial or seed earthquake is specified the subsequent earthquakes are obtained from the statistical distributions of magnitude time and location the magnitude scaling is based on a combination of the gutenberg richter scaling relation and the modified bath s law for the scaling relation of aftershocks relative to the magnitude of the seed earthquake omori s law specifies the distribution of earthquake times and a modified form of omori s law specifies the distribution of earthquake locations since the bass model is specified by the four scaling relations it is fully self similar this is not the case for etas we also give a deterministic version of bass and show that it satisfies tokunaga side branching statistics in a similar way to diffusion limited aggregation dla

integral global optimization method for differential games with application to pursuit evasion games abstract a variation free method for global solution of differential games is proposed the method is based on the integral global optimization algorithm and it does not employ variation based techniques nor the notion of convexity the existence of a saddle point is not assumed a priori instead the method delivers the global minimax and maximin solutions for both players a necessary and sufficient global minimax condition is presented an application is made to the solution of nonlinear pursuit evasion games the ideas are illustrated by examples

performance sensitivity of helicopter global and local optimal harmonic vibration controller a comparison of local and global controllers is conducted for helicopter vibration reduction using actively controlled single dual trailing edge flaps it is found that for high speed flight conditions both local and global controllers perform equally well however for low speed flight local controller gives a marginally better optimal solution compared to the global controller but at the cost of much greater computation time an objective function containing the hub vibratory loads is minimized using the global controller the control effort is weighted such that each flap is used to complete authority a reduction of 73 and 75 in hub vibration is obtained at high speed flight using single and dual flap configurations respectively at low speed flight single and dual flap configurations gave 52 and 63 reduction in vibration respectively numerical studies using an aeroelastic simulation show that the vibration control system performs well up to 5 degrees error in flap phasing from optimal phase the control law is insensitive to small perturbations up to 0 3 deg in the optimal flap harmonics and is fairly insensitive to stiffness and mass changes of the rotor blade as well as noise in the measured hub load data dual flap configuration was found to be more insensitive to errors in optimal flap control harmonics than single flap configuration

predictability of the coherent noise model and its applications we study the threshold distribution function of the coherent noise model for the case of infinite number of agents this function is piecewise constant with a finite number of steps n the latter exhibits a 1 f behavior as a function of the order of occurrence of an avalanche and hence versus natural time an analytic expression of the expectation value mathcal e s for the size s of the next avalanche is obtained and used for the prediction of the next avalanche apart from mathcal e s the number of steps n can also serve for this purpose this enables the construction of a similar prediction scheme which can be applied to real earthquake aftershock data

discontinuous control of nonholonomic systems the problem of asymptotic convergence for a class of nonholonomic control systems via discontinuous control is addressed and solved from a new point of view it is shown that control laws which ensures asymptotic exponential convergence of the closed loop system can be easily designed if the system is described in proper coordinates in such coordinates the system is discontinuous hence the problem of local asymptotic stabilization for a class of discontinuous nonholonomic control systems is dealt with and a general procedure to transform a continuous system into a discontinuous one is presented moreover a general strategy to design discontinuous control laws yielding asymptotic convergence for a class of nonholonomic control systems is proposed the enclosed simulation results show the effectiveness of the method

the relationship between the maximum principle and dynamic programming let v t x be the infimum cost of an optimal control problem viewed as a function of the initial time and state t x dynamic programming is concerned with the properties of v cdot cdot and in particular with its characterization as a solution to the hamilton jacobi bellman equation heuristic arguments have long been advanced relating the maximum principle to dynamic programming according to p t v x left t x 0 t right here x 0 cdot is the minimizing state function under consideration and p cdot is the costate function of the maximum principle in this paper we examine the validity of such claims and find that this relationship interpreted as a differential inclusion involving the generalized gradient is indeed true almost everywhere and at the endpoints for a very large class of nonsmooth optimal control problems

control and stabilization of nonholonomic dynamic systems a class of inherently nonlinear control problems has been identified the nonlinear features arising directly from physical assumptions about constraints on the motion of a mechanical system models are presented for mechanical systems with nonholonomic constraints represented both by differential algebraic equations and by reduced state equations control issues for this class of systems are studied and a number of fundamental results are derived although a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested

boundary regularity of reachable sets of control systems abstract the reachable set of a differential inclusion has nonsmooth boundary in general now we give sufficient conditions on the compact initial set and the differential inclusion x t f t x t u t a e so that this topological boundary has positive erosion i e it satisfies a uniform interior sphere condition the proof is based on the close relation between the semiconcave value function of a suitable mayer problem and its epigraph with positive erosion

universal local versus unified global scaling laws in the statistics of seismicity the unified scaling law for earthquakes proposed by bak christensen danon and scanlon is shown to hold worldwide as well as for areas as diverse as japan new zealand spain or new madrid the scaling functions that account for the rescaled recurrence time probability densities show a power law behavior for long times with a universal exponent about minus 2 2 another decreasing power law governs short times but with an exponent that may change from one area to another this is in contrast with a local time homogenized version of bak et al s procedure which seems to present a universal scaling behavior

adaptive tracking control of uncertain nonholonomic dynamic system this note considers the tracking problem of nonholonomic dynamic systems with unknown inertia parameters a new controller is proposed relying on newly defined tracking errors and the passivity property of the nonholonomic dynamic system the proposed controller ensures that the entire state of the system asymptotically tracks the desired trajectory simulation results show effectiveness of the proposed controller

sensitivity interpretations of the costate variable for optimal control problems with state constraints in optimal control theory it is well known that the costate arc and the associated maximized hamiltonian function can be interpreted in terms of gradients of the value function evaluated along the optimal state trajectory such relations have been referred to as sensitivity relations in the literature we provide in this paper new sensitivity relations for state constrained optimal control problems for the class of optimal control problems considered there is no guarantee that the costate arc is unique a key feature of the results is that they assert some choice of costate arc can be made for which the sensitivity relations are valid the proof technique is to introduce an auxiliary optimal control problem that possesses a richer set of control variables than the original problem the introduction of the additional control variables in effect enlarges the class of variations with respect to which the state trajectory under consideration is a minimizer the extra information thereby obtained yields the desired set of sensitivity relations

global optimization over general compact sets by the beta algorithm a maple code a variant of the beta algorithm based on the cubic algorithm 1 2 is presented for global optimization of holder continuous functions over general compact sets the set monotonic algorithm contains a block for problems with equality constraints and operates within the unit cube 0 1 n universal for all problems on this basis a maple code of modular structure is developed for full global optimization of functions of n variables the code does not create ill conditioned situations graphics are included and the solution set can be visualized in plane projections and sections the code is ready for engineering applications results of numerical experiments are presented with graphs to illustrate the use of the code

reflections on optimality and dynamic programming discrete models and continuous control systems are considered in regard to optimality of their trajectories some aspects of the principle of optimality 1 p 83 are analyzed and it is shown to imply total optimality that is the optimality of every part of an optimal trajectory certain autonomous systems with free admissible variations possess this property nonautonomous optimal systems are not in general totally optimal in which case the principle of optimality is not valid a modification is proposed for the derivation of the main functional equation to demonstrate that dynamic programming and its functional equations are valid also in the case of nonoptimal remaining trajectories under a certain contiguity condition that is defined and analyzed in the paper control systems with incomplete information or structural limitations on controls do not in general satisfy the contiguity condition control problems for such systems may have optimal solutions which however cannot be obtained by dynamic programming this fact is shown in an example of a widely used engineering system for which an optimal trajectory has all its remaining parts nonoptimal and noncontiguous to the optimal trajectory the paper presents theoretical justification of dynamic programming for contiguous systems that do not conform to the principle of optimality examples are presented to illustrate the results which open new avenues in modeling and optimization of general not totally optimal control systems

brief a polytopic game we consider a discrete time linear dynamic game in state space form where the actions of two players are confined to belong to prescribed convex polytopes at all times each player has a home set a polyhedral domain in state space and should devise a cooperative or a non cooperative strategy contingent on the known initial state to reach her own home set in a finite number of steps and stay there irrespective of the action taken by the opponent players act independently on the basis of the state at each time using feedback strategies purpose of the analysis is to classify the set of initial states into winning sets and draw sets for each player and prescribe strategies able to implement the kind of invariance home sets should enjoy in the paper necessary and sufficient conditions for the existence of invariant sets with the above properties are derived these conditions and the relative strategies are characterized in terms of linear programming problems that provide a straightforward algorithmic solution to the game

g leitmann s integral transformation for global optimal control abstract back in 1967 george leitmann has published a note on absolute extrema of certain integrals 1 which later generated much interest in the control systems theory and in effect started a new method for global optimal control in this short paper some observations extensions further developments and perspectives of g leitmann s equivalence transformation are discussed to mark his exceptional contribution to control theory and to facilitate further research in this area following the bright idea of george leitmann s basic lemma on translational symmetry of certain integrals

coextremals and the value function for control problems with data measurable in time abstract suppose x is a solution to an optimal control problem formulated in terms of a differential inclusion known first order necessary conditions of optimality assert existence of a coextremal or adjoint function p which together with x satisfies the hamiltonian inclusion and associated transversality condition in this paper we interpret extremals in terms of generalized gradients of the value function v by demonstrating that p can in addition be chosen to satisfy p t x t p t v t x t a e the hypothesis imposed are more or less the weakest under which the hamiltonian inclusion condition is known to apply and permit in particular measurable time dependence of the data the proof of the results relies on recent developments in hamilton jacobi theory applicable in such circumstances an analogous result is proved for problems where the dynamics are modelled by a differential equation with control term

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

missing data in aftershock sequences explaining the deviations from scaling laws in this paper we extend the branching aftershock sequence model to study the role of missing data at short times and small amplitudes after a mainshock we apply this model which contains three parameters characterizing the missing data to the magnitude and temporal statistics of four aftershock sequences in california we find that the observed time dependent deviations of the frequency magnitude scaling from the gutenberg richter power law dependency can be described quantitatively by the model we also show that for the same set of parameters the model is able to explain quantitatively the observed magnitude dependent deviations of the temporal decay of aftershocks from omori s law in addition we show that the same sets of data can also reproduce quite well the various functional forms of the probability density functions of the return times between consecutive events with magnitudes above a prescribed threshold as well as the violation of scaling at short and intermediate time scales

on the existence of a continuous storage function for dissipative systems conditions for nonlocal existence of a continuous storage function for nonlinear dissipative system are presented more precisely it is shown that under the local w uniform reachability assumption at one point x the required supply function is continuous on the set of points reachable from x conditions for the local w uniform reachability based on the local controllability properties of the system are provided

second order sensitivity relations and regularity of the value function for mayer s problem in optimal control this paper investigates the value function v of a mayer optimal control problem with the state equation given by a differential inclusion first we obtain an invariance property for the proximal and frechet subdifferentials of v along optimal trajectories then we extend the analysis to the sub and superjets of v obtaining new sensitivity relations of second order by applying sensitivity analysis to exclude the presence of conjugate points we deduce that the value function is twice differentiable along any optimal trajectory starting at a point at which v is proximally subdifferentiable we also provide sufficient conditions for the local c 2 regularity of v on neighborhoods of optimal trajectories

global stabilizing near optimal control design for nonholonomic chained systems it is well known that a nonlinear optimal control requires the solution to a two point boundary value problem or to a nonlinear partial differential equation and that such a solution can only be obtained off line by numerical iteration in this paper a new and near optimal control design framework is proposed for controlling any nonholonomic system in the chained form the proposed design is based upon thorough study of uniform complete controllability of the corresponding linear time varying nominal system it is shown that no matter whether the first component u1d t of reference input vector is uniformly nonvanishing or simply nonconvergent to zero or vanishing or identically zero uniform complete controllability of the nominal system can be recovered by employing the proposed time folding unfolding technique upon establishing the common property of uniform complete controllability the proposed framework can be used to design both trajectory tracking control and regulation control in a systematic and unified manner using duality uniform complete observability can also be established a closed form and exponentially convergent observer can be synthesized and the controls designed using the proposed framework can be either state feedback or output feedback the tracking controls are designed using the same three step process that is design of the proposed controls starts with optimal control solutions to two linear nominal subsystems one time invariant and the other time varying the two solutions combined together render a globally stabilizing suboptimal control for the overall system then the optimality condition is invoked to determine the distance between the suboptimal control and the optimal one consequently an improved control can be obtained by introducing a nonlinear additive control term in such a way that the distance aforementioned is minimized as much as possible in closed form an example is used to show that regulation control can be designed similarly all the controls designed are in simple closed forms and hence real time implementable they are time varying and smooth globally and exponentially asymptotically stabilizing and they are near optimal since their closeness to the optimal control attainable only offline can be measured monitored on line and has been minimized

control problems in super articulated mechanical systems control of super articulated mechanical systems sams controlled mechanical system in which the dimension of the configuration space exceeds the dimension of the control input space is studied as a starting point a graphical characterization of general sams is developed in which a so called control flow diagram cfd is constructed to represent the interaction forces among the degrees of freedom of the system three types of structure namely the chain structure the tree structure and isolated points are identified degree of complexity of the system is defined based on the structure of the system a systematic backstepping type controller design is proposed for systems with a chain structure to achieve global asymptotic stability and trajectory tracking such a controller design itself illustrates the sufficient conditions for the existence of smooth feedback control such that the systems with a chain structure can be globally asymptotically stabilized the connection of the authors control algorithm with existing results is discussed

nonextensivity and natural time the case of seismicity nonextensive statistical mechanics pioneered by tsallis has recently achieved a generalization of the gutenberg richter law for earthquakes this remarkable generalization is combined here with natural time analysis which enables the distinction of two origins of self similarity i e the process memory and the process increments infinite variance by using also detrended fluctuation analysis for the detection of long range temporal correlations we demonstrate the existence of both temporal and magnitude correlations in real seismic data of california and japan natural time analysis reveals that the nonextensivity parameter q in contrast to some published claims cannot be considered as a measure of temporal organization but the tsallis formulation does achieve a satisfactory description of real seismic data for japan for q 1 66 when supplemented by long range temporal correlations

conditions for the existence of continuous storage functions for nonlinear dissipative systems the problem of existence of continuous storage functions for dissipative nonlinear systems is considered it is shown that if a nonlinear system is dissipative in the state x then under certain assumptions a continuous storage function can be constructedon a set of points accessible from x by concatenation of a finite number of forwardandbackwardmotions of the system most of these assumptions are weaker than certain controllability type properties andcan be checkedusing similar tests 2004 elsevier b v all rights reserved