on singular control problems with state constraints and regime switching this paper investigates a singular stochastic control problem for a multi dimensional regime switching diffusion process confined in an unbounded domain the objective is to maximize the total expected discounted rewards from exerting the singular control such a formulation stems from application areas such as optimal harvesting multiple species and optimal dividends payments schemes in random environments with the aid of weak dynamic programming principle and an exponential transformation we characterize the value function to be the unique constrained viscosity solution of a certain system of coupled nonlinear quasi variational inequalities several examples are analyzed in detail to demonstrate the main results

the dynamics of repeated impacts with a sinusoidally vibrating table a deceptively simple difference equation is derived which approximately describes the motion of a small ball bouncing vertically on a massive sinusoidally vibrating plate in the case of perfect elastic impacts the equation reduces to the standard mapping which has been extensively studied by physicists in connection with the motions of particles constrained in potential wells it is shown that for sufficiently large excitation velocities and a coefficient of restitution close to one this deterministic dynamical system exhibits large families of irregular non periodic solutions in addition to the expected harmonic and subharmonic motions the physical significance of these and other chaotic motions which appear to occur frequently in non linear oscillations is discussed

minimal positive realizations of transfer functions with nonnegative multiple poles this note concerns a particular case of the minimality problem in positive system theory a standard result in linear system theory states that any nth order rational transfer function of a discrete time invariant linear single input single output siso system admits a realization of order n in some applications however one is restricted to realizations with nonnegative entries i e a positive system and it is known that this restriction may force the order n of realizations to be strictly larger than n a general solution to the minimality problem i e determining the smallest possible value of n is not known in this note we consider the case of transfer functions with nonnegative multiple poles and give sufficient conditions for the existence of positive realizations of order n n with the help of our results we also give an improvement of an existing result in positive system theory

analysis and computation of a discrete costly observation model for growth estimation and management of biological resources abstract early estimation of biological growth of organisms is an indispensable task in ecology and related research areas the biological growth is always time continuous while our observations of the phenomenon are time discrete in practice the formalism of the discrete costly observation dco enables us to mathematically bridge the two qualitatively different processes this formalism is still germinating and its practical applications have not been carried out this paper presents a first application of the dco formalism to a cost effective early estimation problem of the biological growth and its mathematical and numerical analysis growth dynamics of organisms which are fishery resources in this paper is governed by a stochastic differential equation whose solution is observed discretely the optimality equation to be solved for finding the most cost effective observation policy is derived as a fixed point problem based on degenerate parabolic partial differential equations the fixed point problem turns out to be uniquely solvable a recursive approximation of the fixed point problem is presented and its solvability in a viscosity sense is discussed a finite different scheme is then employed to fully discretize the recursive equations the present model is finally applied to a problem of japanese smelt plecoglossus altivelis altivelis ayu an important inland fishery resource in japan

on optimal harvesting problems in random environments this paper investigates the optimal harvesting strategy for a single species living in random environments whose population growth is given by a regime switching diffusion harvesting acts as a stochastic control on the size of the population the objective is to find a harvesting strategy which maximizes the expected total discounted income from harvesting up to the time of extinction of the species the income rate is allowed to be state and environment dependent this is a singular stochastic control problem with both the extinction time and the optimal harvesting policy depending on the initial condition one aspect of receiving payments up to the random time of extinction is that small changes in the initial population size may significantly alter the extinction time when using the same harvesting policy consequently one no longer obtains continuity of the value function using standard arguments for either regular or singular control problems having a fixed time horizon this paper introduces a new sufficient condition under which the continuity of the value function for the regime switching model is established further it is shown that the value function is a viscosity solution of a coupled system of quasi variational inequalities the paper also establishes a verification theorem and based on this theorem an varepsilon optimal harvesting strategy is constructed under certain conditions on the model two examples are analyzed in detail

numerical methods for optimal dividend payment and investment strategies of regime switching jump diffusion models with capital injections this work focuses on numerical methods for finding optimal investment dividend payment and capital injection policies to maximize the present value of the difference between the cumulative dividend payment and the possible capital injections the surplus is modeled by a regime switching jump diffusion process subject to both regular and singular controls using the dynamic programming principle the value function is a solution of the coupled system of nonlinear integro differential quasi variational inequalities in this paper the state constraint of the impulsive control gives rise to a capital injection region with free boundary which makes the problem even more difficult to analyze together with the regular control and regime switching the closed form solutions are virtually impossible to obtain we use markov chain approximation techniques to construct a discrete time controlled markov chain to approximate the value function and optimal controls convergence of the approximation algorithms is proved examples are presented to illustrate the applicability of the numerical methods

an efficient algorithm for positive realizations we observe that successive applications of known results from the theory of positive systems lead to an it efficient general algorithm for positive realizations of transfer functions we give two examples to illustrate the algorithm one of which complements an earlier result of cite large finally we improve a lower bound of cite mn2 to indicate that the algorithm is indeed efficient in general

a tutorial on the positive realization problem this paper is a tutorial on the positive realization problem that is the problem of finding a positive state space representation of a given transfer function and characterizing existence and minimality of such representation this problem goes back to the 1950s and was first related to the identifiability problem for hidden markov models then to the determination of internal structures for compartmental systems and later embedded in the more general framework of positive systems theory within this framework developing some ideas sprang in the 1960s during the 1980s the positive realization problem was reformulated in terms of a geometric condition which was recently exploited as a tool for finding the solution to the existence problem and providing partial answers to the minimality problem in this paper the reader is carried through the key ideas which have proved to be useful in order to tackle this problem in order to illustrate the main results contributions and open problems several motivating examples and a comprehensive bibliography on positive systems organized by topics are provided

a new approach to the internally positive representation of linear mimo systems the problem of representing linear systems through combination of positive systems is relevant when signal processing schemes such as filters state observers or control laws are to be implemented using positive technologies such as charge routing networks and fiber optic filters this problem well investigated in the siso case can be recasted into the more general problem of internally positive representation ipr of systems this paper presents a methodology for the construction of such iprs for mimo systems based on a suitable convex positive representation of complex vectors and matrices the stability properties of the iprs are investigated in depth achieving the important result that any stable system admits a stable ipr of finite dimension a main algorithm and three variants all based on the proposed methodology are presented for the construction of stable iprs all of them are straightforward and are characterized by a very low computational cost the first and second may require a large state space dimension to provide a stable ipr while the third and the fourth are aimed to provide stable iprs of reduced order

a positivity based approach to delay dependent stability of systems with large time varying delays abstract in this paper we propose new explicit tests for positivity and exponential stability of systems with large time varying delays our approach is based on nonoscillation of solutions of the corresponding diagonal scalar delay differential equations numerical examples illustrate the efficiency of the results

optimal harvesting strategies for stochastic competitive lotka volterra ecosystems this work develops optimal harvest strategies for lotka volterra systems so as to establish economically ecologically and environmentally reasonable strategies for populations subject to the risk of extinction to better reflect reality a continuous time markov chain is used to model the random environment the underlying systems are thus controlled regime switching diffusions that belong to the class of singular control problems starting with a model having multiple species we construct upper bounds for the value functions prove the finiteness of the harvesting value and derive properties of the value functions then we construct explicit chattering harvesting strategies and the corresponding lower bounds for the value functions by using the idea of harvesting only one species at a time we further show that this is a reasonable candidate for the best lower bound that one can expect moreover in some cases the lower bounds provide a good approximation of the value functions

convergent numerical scheme for singular stochastic control with state constraints in a portfolio selection problem we consider a singular stochastic control problem with state constraints that arises in problems of optimal consumption and investment under transaction costs numerical approximations for the value function using the markov chain approximation method of kushner and dupuis are studied the main result of the paper shows that the value function of the markov decision problem mdp corresponding to the approximating controlled markov chain converges to that of the original stochastic control problem as various parameters in the approximation approach suitable limits all our convergence arguments are probabilistic the main assumption that we make is that the value function be finite and continuous in particular uniqueness of the solutions of the associated hjb equations is neither needed nor available in the generality under which the problem is considered specific features of the problem that make the convergence analysis nontrivial include unboundedness of the state and control space and the cost function degeneracies in the dynamics mixed boundary dirichlet neumann conditions and presence of both singular and absolutely continuous controls in the dynamics finally schemes for computing the value function and optimal control policies for the mdp are presented and illustrated with a numerical study

discontinuity mappings for stochastic nonsmooth systems abstract for stability and bifurcation analysis involving recurrent behaviour such as periodic orbits it is important to be able to quantify how nearby trajectories behave by means of a local mapping in smooth systems these mappings can be computed using the system s variational equations for piecewise smooth or hybrid systems the same technique cannot be used without some corrections this is due to the fact that nearby trajectories can be topologically distinct because they can undergo different sequences of events associated with the system s discontinuity boundaries to account for this one can derive zero time discontinuity mappings associated with boundary interactions in this paper we derive zero time discontinuity mappings for piecewise smooth vector fields and hybrid dynamical systems in which the position of the discontinuity boundary has a stochastic component in particular we consider systems with stochastically oscillating boundaries and systems with stochastic imperfections on the discontinuity boundary

frequency domain limitations in the design of nonnegative impulse response filters this paper provides a fundamental characterization of nonnegative impulse response nnir filters the analysis is performed in terms of both geometrically and equidistantly spaced frequencies various inferences on the frequency domain limitations in nnir filter design are analyzed it is found that a performance boundary exists for nnir lowpass filters independent of the system order a specification test procedure is also provided for nnir lowpass filters based on these analyses the results are presented for systems in the discrete time d t domain although equivalent results exist for systems in the continuous time domain

numerical solutions of optimal risk control and dividend optimization policies under a generalized singular control formulation this paper develops numerical methods for finding optimal dividend pay out and reinsurance policies a generalized singular control formulation of surplus and discounted payoff function is introduced where the surplus is modeled by a regime switching process subject to both regular and singular controls to approximate the value function and optimal controls markov chain approximation techniques are used to construct a discrete time controlled markov chain the proofs of the convergence of the approximation sequence to the surplus process and the value function are given examples of proportional and excess of loss reinsurance are presented to illustrate the applicability of numerical methods

noise and multistability in the square root map abstract in this paper we describe the complex structure of the basins of attraction of stable periodic orbits of the one dimensional square root map and how this produces sensitivity to the addition of small amplitude noise in particular we focus on how noise of varying amplitudes affects the system in parameter regions of attractor coexistence and also how trajectories jump between different periodic behaviours we show that there is a non monotonic relationship between the noise amplitude and the proportion of time spent in each periodic behaviour these relationships are explained by examining approximations of steady state distributions of trajectory deviations due to noise and the complicated deterministic structures of the map we also show how the effect of noise scales on consecutive intervals of multistability

order bound for the realization of a combination of positive filters in a problem on the realization of digital filters initiated by gersho and gopinath we extend and complete a remarkable result of benvenuti farina and anderson on decomposing the transfer function t z of an arbitrary linear asymptotically stable discrete time invariant single input single output system as a difference t z t1 z t2 z of two positive asymptotically stable linear systems we give an easy to compute algorithm to handle the general problem in particular also the case of transfer functions t z with multiple poles which was left open in a previous paper one of the appearing positive asymptotically stable systems is always one dimensional while the other has dimension depending on the order and in the case of nonreal poles also on the location of the poles of t z the appearing dimension is seen to be minimal in some cases and it can always be calculated before carrying out the realization

a discontinuity geometry view of the relationship between saddle node and grazing bifurcations abstract in this study we will introduce a topological approach termed as discontinuity geometry for both analysis and visualization of the dynamics of periodically forced impact oscillators after a brief outline of the methodology and the general representation of trajectories with impacts the framework will be used to analyse grazing bifurcations in particular it will be used to geometrically explain why there is sometimes a saddle node bifurcation present in the vicinity of grazing bifurcations and why it is sometimes absent

a construction method for positive realizations with an order bound abstract this paper presents an efficient construction method to address the positive realization problem with an order bound for primitive transfer functions with multiple real or complex conjugate pairs of poles using polyhedral cone concepts through a down sampling step the non primitive transfer function is decomposed into a sum of primitive transfer functions a construction method for the positive realization of a primitive transfer function with multiple poles is proposed by using a regular polygon in the complex plane a sufficient condition for the existence of the positive realization with an order bound is analyzed numerical examples are provided to demonstrate the efficiency of the proposed method for obtaining positive realizations

routing and singular control for queueing networks in heavy traffic the problem of routing control in an open queueing network under conditions of heavy traffic and finite scaled buffers is dealt with the operating statistics can be state dependent the sequence of scaled controlled state processes converges to a singularly controlled reflected diffusion with the associated costs under broad conditions due to the nature of the controls a scaling method is introduced to obtain the convergence since the actual sequence of processes does not necessarily converge in the skorokhod topology owing to finite buffers an extension of the reflection mapping needs to be obtained the optimal value functions for the physical processes converge to the optimal value function of the limit process under broad conditions approximations to the optimal control for the limit process are obtained as well as properties of the sequence of physical processes the optimal or controlled but not necessarily optimal limit process can be used to approximate a large variety of functio

bifurcations in nonsmooth dynamical systems a review is presented of the one parameter nonsmooth bifurcations that occur in a variety of continuous time piecewise smooth dynamical systems motivated by applications a pragmatic approach is taken to defining a discontinuity induced bifurcation dib as a nontrivial interaction of a limit set with respect to a codimension one discontinuity boundary in phase space only dibs that are local are considered that is bifurcations involving equilibria or a single point of boundary interaction along a limit cycle for flows three classes of systems are considered involving either state jumps jumps in the vector field or jumps in some derivative of the vector field a rich array of dynamics are revealed involving the sudden creation or disappearance of attractors jumps to chaos bifurcation diagrams with sharp corners and cascades of period adding for each kind of bifurcation identified where possible a kind of normal form or discontinuity mapping dm is given together with a canonical example and an application the goal is always to explain dynamics that may be observed in simulations of systems which include friction oscillators impact oscillators dc dc converters and problems in control theory

stability analysis for time varying systems with delay using linear lyapunov functionals and a positive systems approach we prove stability of time varying systems with delays using linear lyapunov functionals and positive systems and we provide robustness of the stability with respect to multiplicative uncertainty in the vector fields we allow cases where the delay may be unknown and where the vector fields defining the systems are not necessarily bounded we illustrate our work using a chain of integrators and other examples

numerical methods for stochastic singular control problems the paper develops a powerful class of numerical methods for stochastic singular control problems the basic models used are diffusion or reflected diffusions but the method is of general applicability the central idea is that of the markov chain approximation method where an approximation to the control problem is found for which an optimal solution is computable and which is an arbitrarily good approximation to the original problem and its optimal value function the methods are convenient to program and use and they have been used with success and they cover a wide variety of problems in fact for the singular problem they seem to be the only ones currently available owing to problems in proving tightness of certain processes that occur in the convergence proofs the methods of proof used for the nonsingular problems need modifications examples of useful approximations the algorithms and the convergence proofs are given to illustrate the power of the methods two classes of problems are d

numerical methods for optimal harvesting strategies in random environments under partial observations this work is concerned with optimal harvesting problems in random environments in contrast to the existing literature the markov chain is hidden and can only be observed in a gaussian white noise in our work we first use the wonham filter to estimate the markov chain from the observable evolution of the given process so as to convert the original problem to a completely observable one then we treat the resulting optimal control problem because the problem is virtually impossible to solve in closed form our main effort is devoted to developing numerical approximation algorithms to approximate the value function and optimal strategies markov chain approximation methods are used to construct a discrete time controlled markov chain convergence of the algorithm is proved by weak convergence method and suitable scaling a numerical example is provided to demonstrate the results

internally positive representation of a class of continuous time systems the problem of realizing arbitrary transfer functions as combinations of positive filters or more in general of representing arbitrary systems by means of internally positive representations iprs has been widely investigated in the discrete time framework an ipr is a positive state space representation endowed with input state and output transformations that realizes arbitrary input state output dynamics this technical note investigates the problem of the ipr construction for continuous time systems and proposes a technique that provides stable iprs for a particular class of systems

non periodic motion caused by grazing incidence in an impact oscillator the motion of a single degree of freedom periodically forced oscillator subjected to a rigid amplitude constraint is considered using analytical methods the singularities caused by grazing impact are studied it is shown that as a stable periodic orbit comes to grazing impact under the control of a single parameter a special type of bifurcation occurs the motion after the bifurcation may be non periodic and a criterion for this based on orientation and eigenvalues is given

bifurcations caused by grazing incidence in many degrees of freedom impact oscillators the transition from stable periodic non impacting motion to impacting motion is analysed for a mechanical oscillator by using local methods it is shown that a grazing impact leads to an almost one dimensional stretching in state space a condition can then be formulated such that a grazing trajectory will be stable if the condition is fulfilled if this is the case the bifurcation will be continuous and the motion after the bifurcation can be understood by a one dimensional mapping this mapping is known to exhibit chaotic solutions as well as arbitrary long stable cycles depending on parameters

a diffusion model for optimal dividend distribution for a company with constraints on risk control this paper investigates a model of a corporation which faces constant liability payments and which can choose a production business policy from an available set of control policies with different expected profits and risks the objective is to find a business policy and a dividend distribution scheme so as to maximize the expected present value of the total dividend distributions the main feature of this paper is that there are constraints on business activities such as inability to completely eliminate risk even at the expense of reducing the potential profit to zero or when such a risk cannot exceed a certain level the case in which there is no restriction on the dividend pay out rates is dealt with this gives rise to a mixed regular singular stochastic control problem first the value function is analyzed in great detail and in particular is shown to be a viscosity solution of the corresponding hamilton jacobi bellman hjb equation based on this it is further proved that the value function must be twice continuously differentiable then a delicate analysis is carried out on the hjb equation leading to an explicit expression of the value function as well as the optimal policies