optimization of a spatially variable resource an illustration for irrigated crops economic optimization of a resource which is space dependent and displays irregular spatial variation was investigated the resource yield of an entire field is regarded as a random variable which depends on a random function on the one hand and a control variable on the other hand the study focuses on the question of how economically efficient levels of relevant control variables may vary as a function of random spatial variability of resources taking into account the decision maker s attitude to risk a general approach to cope with the above question is developed and the effects of uncertainty on the levels of the optimal control variables are investigated by comparing a deterministic case with two uncertainty levels of decision making risk neutral and risk averse following the general methodology expectation value and variance of average yield are first approximated by taylor series expansion and then closed form solutions are obtained for a specific exponential yield production function subsequently we illustrate the general analysis for the example of irrigated corn the only control variable being water quantity and with corp yield a spatial random function in the plane the results suggest that the impact of random soil properties on optimum level of water application might be substantial hence it is generally worthwhile to account for spatial variability in optimizing level of irrigation or similar control variables

controllability and observability for a class of controlled switching impulsive systems in this note we study the controllability and observability problem for a class of controlled switching impulsive systems by proposing several formula of variation of parameters for this type of time varying systems and employing the characteristic polynomial theory of matrix we establish necessary and sufficient conditions for controllability and controlled observability with respect to a given switching time sequence specializing the obtained results to the case of time invariant linear switching impulsive systems we derive some simple algebraic criteria which include the results reported in the literature for time invariant linear switching systems linear impulsive systems and classical linear systems one example is worked out for illustration

dynamic programming for nonlinear systems driven by ordinaryand impulsive controls a dynamic programming approach is considered for a class of minimum problems with impulses the minimization domain consists of trajectories satisfying an ordinary differential equation whose right hand side depends not only on a measurable control v but also on a second control u and on its time derivative dot u for this reason the control u and the differential equation are called impulsive the value function of the considered minimum problem turns out to depend on the time the state the u variable and the variation allowed to the impulsive control it is shown that the value function satisfies in a generalized sense a dynamic programming equation dpe which is obtained from a dynamic programming principle involving space time trajectories moreover the value function is the unique map solving equation dpe satisfying either an inequality condition or a supersolution condition at each point of the boundary incidentally this extends a result by barron jensen and menaldi nonlinear anal 21 1993 pp 241 268 where the impulsive control is scalar monotone and the corresponding vector field is independent of the state variable next a maximum principle is proved and the well known relationship between adjoint variables and value function is suitably extended to impulsive control systems a fully elaborated example concludes the paper

variability of yield of an irrigated crop and its causes 3 numerical simulation and field results the prediction of yield variability of an irrigated crop as influenced by the uncertainty of soil plant and boundary conditions parameters and an investigation through numerical simulation using the probability theory apparatus are tested against field data among the large number 19 of parameters six which have a significant impact on yield variability have been singled out these include two plant parameters with an average relative contribution of 23 three soil hydraulic parameters with a relative impact of 47 and one irrigation boundary condition parameter related to the nonuniformity of water application with 30 average contribution upon yield variability an additional set of results is related to the yield coefficient of variation its range was simulated to be generally between 0 3 and 1 with an average of approximately 0 6 a similar range but with a somewhat lower average was measured in the field the detection of the parameters of large impact obtained by both simulation and field validation give an indication of weak links which require further investigation and for which collection of field data is important

on the regularization of zeno hybrid automata fundamental properties of hybrid automata such as existence and uniqueness of executions are studied particular attention is devoted to zeno hybrid automata which are hybrid automata that take infinitely many discrete transitions in finite time it is shown that regularization techniques can be used to extend the zeno executions of these automata to times beyond the zeno time different types of regularization may however lead to different extensions a water tank control problem and a bouncing ball system are used to illustrate the results

measure driven differential inclusions measure driven differential inclusions arise when we attempt to derive necessary conditions of optimality for optimal impulsive control problems with nonsmooth data we introduce the concept of a robust solution to a measure driven inclusion which extends to a multifunction setting interpretations of solutions to measure driven differential equations provided by dal maso and rampazzo and others closure properties of sets of robust solutions are established and notions of relaxation investigated implications for optimality conditions for impulsive control problems are pursued in a companion paper

linear multistep methods for impulsive delay differential equations abstract this paper deals with the convergence and stability of linear multistep methods for a class of linear impulsive delay differential equations numerical experiments show that the simpson s rule and two step bdf method are of order p 0 when applied to impulsive delay differential equations an improved linear multistep numerical process is proposed convergence and stability conditions of the numerical solutions are given in the paper numerical experiments are given in the end to illustrate the conclusion

solutions to hybrid inclusions via set and graphical convergence with stability theory applications motivated by questions in stability theory for hybrid dynamical systems we establish some fundamental properties of the set of solutions to such systems using the notion of a hybrid time domain and general results on set and graphical convergence we establish under weak regularity and local boundedness assumptions that the set of solutions is sequentially compact and upper semicontinuous with respect to initial conditions and system perturbations the general facts are then used to establish several results for the behavior of hybrid systems that have asymptotically stable compact sets these results parallel what is already known for differential inclusions and difference inclusions for example the basin of attraction for a compact attractor is relatively open the attractivity is uniform from compact subsets of the basin of attraction and asymptotic stability is robust with respect to small perturbations

an analogue of row removal for diagrammatic cherednik algebras we prove an analogue of james donkin row removal theorems for diagrammatic cherednik algebras this is one of the first results concerning the graded decomposition numbers of these algebras over fields of arbitrary characteristic as a special case our results yield a new reduction theorem for graded decomposition numbers and extension groups for cyclotomic q schur algebras

variability of yield of an irrigated crop and its causes 2 input data and illustration of results the data needed in order to evaluate the impact of the variability of different parameters upon crop yield variability were collected from various sources in the literature two fields paoche in the united states and bet dagan in israel which had been studied extensively in the past were selected as representative for a wide range of soil conditions in a similar manner standard values of the expectations and extreme values of variances of other soil plant atmospheric parameters were inferred from the literature the adopted values are summarized in a table the dependence of the relative yield and its derivatives upon each of 19 parameters while the remaining ones are kept fixed at their standard expected value is illustrated

optimality of affine control system of several species in competition on a sequential batch reactor in this paper we analyze the optimalty of affine control system of several species in competition for a single substrate on a sequential batch reactors sbr with the objective being to reach a given low level of the substrate we allow controls to be bounded measurable functions of time plus possible impulses a suitable modification of the dynamics leads to a slightly different optimal control problem without impulsive controls for which we apply different optimality conditions de rived from pontryagin principle and the hamilton jacobi bellman equation we thus characterize the singular trajectories of our problem as the extremal trajecto ries keeping the substrate at a constant level we also establish conditions for which a immediate one impulse ioi strategy is optimal some numerical experiences are then included in order to illustrate our study and show that those conditions are also necessary to ensure the optimality of the ioi strategy

higher order runge kutta methods for impulsive differential systems abstract this paper studies higher order approximations of solutions of differential equations with non fixed times of impulses we assume that the right hand side is sufficiently smooth using a runge kutta method of higher order and natural assumptions on the impulsive surfaces and the impulses we calculate good approximations of the jump times which enables us to extend the classical results for higher order of convergence of runge kutta methods to more complicated systems

discrete approximations and refined euler lagrange conditions fornonconvex differential inclusions this paper deals with the bolza problem p for differential inclusions subject to general endpoint constraints we pursue a twofold goal first we develop a finite difference method for studying p and construct a discrete approximation to p that ensures a strong convergence of optimal solutions second we use this direct method to obtain necessary optimality conditions in a refined euler lagrange form without standard convexity assumptions in general we prove necessary conditions for the so called intermediate relaxed local minimum that takes an intermediate place between the classical concepts of strong and weak minima in the case of a mayer cost functional or boundary solutions to differential inclusions this euler lagrange form holds without any relaxation the results obtained are expressed in terms of nonconvex valued generalized differentiation constructions for nonsmooth mappings and sets

necessary conditions for optimal impulsive control problems necessary conditions of optimality in the form of a maximum principle are derived for a class of optimal control problems certain of whose controls are represented by measures and whose state trajectories are functions of bounded variation state trajectories are interpreted as robust solutions of the dynamic equations a concept of solutions which takes account of the interaction between the measure control and the state variables during the jumps the maximum principle which is derived improves on earlier optimality conditions for problems of this nature by allowing nonsmooth data measurable time dependence and a possibly time varying constraint set for the conventional controls

the algorithmic analysis of hybrid systems we present a general framework for the formal specification and algorithmic analysis of hybrid systems a hybrid system consists of a discrete program with an analog environment we model hybrid systems as finite automata equipped with variables that evolve continuously with time according to dynamical laws for verification purposes we restrict ourselves to linear hybrid systems where all variables follow piecewise linear trajectories we provide decidability and undecidability results for classes of linear hybrid systems and we show that standard program analysis techniques can be adapted to linear hybrid systems in particular we consider symbolic model checking and minimization procedures that are based on the reachability analysis of an infinite state space the procedures iteratively compute state sets that are definable as unions of convex polyhedra in multidimensional real space we also present approximation techniques for dealing with systems for which the iterative procedures do not converge

dynamical properties of hybrid automata hybrid automata provide a language for modeling and analyzing digital and analogue computations in real time systems hybrid automata are studied here from a dynamical systems perspective necessary and sufficient conditions for existence and uniqueness of solutions are derived and a class of hybrid automata whose solutions depend continuously on the initial state is characterized the results on existence uniqueness and continuity serve as a starting point for stability analysis lyapunov s theorem on stability via linearization and lasalle s invariance principle are generalized to hybrid automata

statistical inference of spatial random functions a methodology to infer the spatial distribution of the statistical moments of a stationary normal random spatial function based on a finite set of measured values is described the uncertainty of estimation of the parameters characterizing the probability density function pdf is incorporated in the process of inferring and conditioning relationships generalizing the traditional approach of stochastic interpolation and simulation have been derived they yield i an estimate of the conditional expectation ii an estimate of the conditional multivariate normal mvn covariance and variance and iii a new quantity the variance of the estimated variance the method is illustrated for a two dimensional synthetic example with both estimated mean and variance regarded as random a set of 60 points was generated within a square of sides equal to unity from a mvn population with zero mean unity variance and exponential covariance with a linear integral scale of 0 3 contour maps of the inferred expected values and variances are drawn and compared with maps of kriged values and maps of kriged variances contour maps of variances of estimation of the variance are given to demonstrate the uncertainty of variance estimates the unconditional and conditional statistical moments of the space average block value were also evaluated the reduction of the uncertainty of the inferred random variable as the number of measured points increases is demonstrated with the same synthetic example

on reachability and minimum cost optimal control questions of reachability for continuous and hybrid systems can be formulated as optimal control or game theory problems whose solution can be characterized using variants of the hamilton jacobi bellman or isaacs partial differential equations the formal link between the solution to the partial differential equation and the reachability problem is usually established in the framework of viscosity solutions this paper establishes such a link between reachability viability and invariance problems and viscosity solutions of a special form of the hamilton jacobi equation this equation is developed to address optimal control problems where the cost function is the minimum of a function of the state over a specified horizon the main advantage of the proposed approach is that the properties of the value function uniform continuity and the form of the partial differential equation standard hamilton jacobi form continuity of the hamiltonian and simple boundary conditions make the numerical solution of the problem much simpler than other approaches proposed in the literature this fact is demonstrated by applying our approach to a reachability problem that arises in flight control and using numerical tools to compute the solution

lack of bv bounds for impulsive control systems abstract for a nonlinear impulsive control system we extend the so called graph completion approach and introduce a notion of generalized solution x associated to a control u whose total variation is bounded on 0 t for every t t but possibly unbounded on 0 t we prove existence consistency with classical solutions and well posedness of this solution in particular we characterize it as a pointwise limit of certain regular solutions the notion that we consider provides the natural setting for controllability questions and for some non coercive optimal control problems where chattering phenomena at the final time are expected more in general it is well suited to describe the evolution of control systems subject to a train of impulses where no a priori bounds on the number and the amplitude of the impulses are imposed

l1 limit solutions for control systems abstract for a control cauchy problem x f t x u v 1 m g x u x a x on an interval a b we propose a notion of limit solution x verifying the following properties i x is defined for l 1 impulsive inputs u and for standard bounded measurable controls v ii in the commutative case i e when g g 0 for all 1 m x coincides with the solution one can obtain via a change of coordinates that makes the g simultaneously constant iii x subsumes former concepts of solution valid for the generic noncommutative case in particular when u has bounded variation we investigate the relation between limit solutions and single valued graph completion solutions furthermore we prove consistency with the classical caratheodory solution when u and x are absolutely continuous even though some specific problems are better addressed by means of special representations of the solutions we believe that various theoretical issues call for a unified notion of trajectory for instance this is the case of optimal control problems possibly with state and endpoint constraints for which no extra assumptions like e g coercivity bounded variation commutativity are made in advance

a maximum principle for optimal processes with discontinuous trajectories a maximum principle is proved which governs solutions to dynamic optimization problems in which the controls driving the system may be impulsive and give rise to discontinuous trajectories the approach which involves approximating the problem by a conventional one and using ekeland s theorem is new it permits us to weaken very considerably the hypotheses under which maximum principles for such problems have previously been proved

hamilton jacobi formulation for reach avoid differential games a new framework for formulating reachability problems with competing inputs nonlinear dynamics and state constraints as optimal control problems is developed such reach avoid problems arise in among others the study of safety problems in hybrid systems earlier approaches to reach avoid computations are either restricted to linear systems or face numerical difficulties due to possible discontinuities in the hamiltonian of the optimal control problem the main advantage of the approach proposed in this paper is that it can be applied to a general class of target hitting continuous dynamic games with nonlinear dynamics and has very good properties in terms of its numerical solution since the value function and the hamiltonian of the system are both continuous the performance of the proposed method is demonstrated by applying it to a case study which involves the target hitting problem of an underactuated underwater vehicle in the presence of obstacles

minimal time sequential batch reactors with bounded and impulse controls for one or more species we consider the optimal control problem of feeding in minimal time a tank where several species compete for a single resource with the objective being to reach a given level of the resource we allow controls to be bounded measurable functions of time plus possible impulses for the one species case we show that the immediate one impulse strategy filling the whole reactor with one single impulse at the initial time is optimal when the growth function is monotonic for nonmonotonic growth functions with one maximum we show that a particular singular arc strategy precisely defined in section 3 is optimal these results extend and improve former ones obtained for the class of measurable controls only for the two species case with monotonic growth functions we give conditions under which the immediate one impulse strategy is optimal we also give optimality conditions for the singular arc strategy at a level that depends on the initial condition to be optimal the possibility for the immediate one impulse strategy to be nonoptimal while both growth functions are monotonic is a surprising result and is illustrated with the help of numerical simulations

unsaturated flow in spatially variable fields 2 application of water flow models to various fields the method of modeling water flow during infiltration and redistribution formulated in part 1 of this study has been applied to compute expectations and variances of a few water flow variables and of effective hydraulic properties two spatially variable soils with different degrees of variation have been investigated the expectations and variances are obtained by using the statistical averaging procedure and probability density function pdf of saturated hydraulic conductivity ks the stationarity hypotheses and the requirement that the integral scale of ks is much smaller than the length scale characterizing the field in the x y plane have been adopted an approximate solution which assumes the concept of a wetting front and uniform water content is used for the statistical averaging procedure a comparison of these results with data computed by a more accurate numerical solution to richard s equation shows that the approximate simplified models lead to a quite accurate value of the expectations and variances of the flow variables when the field is sufficiently heterogeneous it is suggested that in spatially variable fields stochastic modeling represents the actual flow phenomena more realistically and provides the main statistical moments mean variances by using simplified flow models which can be used with confidence in applications the field effective hydraulic properties have been defined and derived by using approximate models it is shown that effective properties may be meaningful only under very restricted and special conditions such as steady gravitational flow they do not exist in the general case of infiltration redistribution it is concluded that the traditional deterministic approach for solving the flow equations cannot be justified for solving flow problems in spatially variable fields

variability of yield of an irrigated crop and its causes 1 statement of the problem and methodology crop yield depends on a large number of soil water and atmosphere related parameters some of these parameters are spatially variable and most of them are subject to uncertainty consequently the yield is also spatially variable and its prediction is uncertain the aims of this study were to 1 estimate the degree of yield variability as a function of variability of various parameters and 2 attempt to single out those parameters the variability of which are estimated to have the largest impact upon the variability of yield of an irrigated crop the first part of the study presents the general methodology developed in order to achieve these aims it consists of a numerical deterministic model which predicts crop yield relative to its potential yield for a given irrigation regime and quality of water and for given soil and atmospheric conditions subsequently the model is used to derive the values of expectation and variance of the relative irrigated crop yield as functions of these values of the various parameters

existence theorems for optimal control and calculus of variations problems where the states can jump we examine optimal control and calculus of variations problems where the states can be discontinuous in the control theory setting this is caused by allowing impulse controls conditions under which optimal solutions will exist for these problems are determined after we derive suitable objective functionals that can handle the jump terms

impulse differential inclusions a viability approach to hybrid systems impulse differential inclusions are introduced as a framework for modeling hybrid phenomena connections to standard problems in the area of hybrid systems are discussed conditions are derived that allow one to determine whether a set of states is viable or invariant under the action of an impulse differential inclusion for sets that violate these conditions methods are developed for approximating their viability and invariance kernels that is the largest subset that is viable or invariant under the action of the impulse differential inclusion the results are demonstrated on examples

a sampling method and approximation results for impulsive systems this paper studies an impulsive dynamical system that is in the form of a measure driven differential inclusion the employed solution concept depends upon a graph completion of the measure the first main result shows that a subsequence of discrete time trajectories graph converges to a solution and under lipschitz hypotheses that every solution can be obtained in this manner the second main result pursues a similar development with classical systems that approximate the given one

unsaturated flow in spatially variable fields 3 solute transport models and their application to two fields an approximate model of salt transport in a spatially variable field during infiltration and redistribution is presented the water flow is assumed to be vertical and both pore water velocity and depth of wetting front change in the horizontal plane due to the variability of soil hydraulic properties these variables are computed with the aid of the model developed in part 1 dagan and bresler this issue and part 2 bresler and dagan this issue of the present study the salt transport in the vertical profile is computed by using an approximate closed form solution of the convection dispersion equation with the value of the dispersivity increasing from zero to its maximum 3 cm as the front propagates downward the concentration profiles vary in the horizontal plane because of the aforementioned variation of water flow variables the expectation value and the variance of the concentration are computed as function of depth and time for the two variable soils considered in part 2 panoche with large variability and bet dagan with lesser variability the results are compared with those based on a numerical simulation of the water flow and salt transport as well as with those pertaining to an equivalent deterministic uniform soil for the expected value it is shown that the approximate model yields quite accurate results when compared with the numerical simulations for the field of large variability while some differences are present for the more uniform one in any case the results for the expected value based on the traditional approach of replacing the variable field by an equivalent uniform one include a much larger error it is suggested that the approximate simplified model presented here may serve as a basis for simulation of additional effects not considered here e g adsorption root extraction and change of soil properties due to salt content the main conclusion is that in spite of the apparent complexity the statistical moments of salt concentration in a spatially variable field can be determined by using simple flow models therefore further refinement of models of transport in homogeneous columns does not seem to be warranted and the research effort should be concentrated on the study of field variability

differential inclusions and target problems where and how solutions associated to a differential inclusion can or cannot enter a given target is studied for this purpose partitions of the target boundary are associated with the dynamic of the system the behaviour of these solutions is qualitatively described in terms of viability and invariance kernels of sets these kernels determine points such that there exist respectively all solutions starting at these points remain in a given set of constraints the sets that are reached in finite time by viable solutions to the system are also studied finally some applications to control systems with one target are provided and the concept of semipermeable barriers will be generalized

sign sequences and decomposition numbers we obtain a closed formula for the v decomposition numbers d v arising from the canonical basis of the fock space representation of uv le where the partition is obtained from by moving some nodes in its young diagram all of which having the same e residue we also show that when these v decomposition numbers are evaluated at v 1 we obtain the corresponding decomposition numbers for the schur algebras and symmetric groups

hamilton jacobi bellman equation and feedback synthesis for impulsive control there is an increasing number of applications whose trajectories are better modeled by discontinuous or impulsive trajectories thus we explore optimality conditions for impulsive control system expressed in terms of an hamilton jacobi bellman equation we use a measure driven differential inclusion to model the impulsive behavior since it provides a formal framework in which the control space is complete additionally we use the impulsive euler solution and invariance results to derive feedback optimal control synthesis

categorifications and cyclotomic rational double affine hecke algebras varagnolo and vasserot conjectured an equivalence between the category mathcal o for crdaha s and a subcategory of an affine parabolic category o of type a we prove this conjecture as applications we prove a conjecture of rouquier on the dimension of simple modules of crdaha s and a conjecture of chuang miyachi on the koszul duality for the category mathcal o of crdaha s

kleshchev s decomposition numbers for diagrammatic cherednik algebras we construct a family of graded isomorphisms between certain subquotients of diagrammatic cherednik algebras as the quantum characteristic multicharge level degree and weighting are allowed to vary this provides new structural information even in the case of the classical q schur algebra this also allows us to prove some of the first results concerning the graded decomposition numbers of these algebras over fields of arbitrary characteristic