on the capacity formula for multiple input multiple output wireless channels a geometric interpretation the capacity of multiple input multiple output mimo wireless channels is computed for ricean channels the novelty is a geometrical ray tracing interpretation of the mimo channel capacity formula to find array geometries which greatly enhance channel capacity compared to single input single output siso systems

mixed integer optimal control under minimum dwell time constraints tailored mixed integer optimal control policies for real world applications usually have to avoid very short successive changes of the active integer control minimum dwell time mdt constraints express this requirement and can be included into the combinatorial integral approximation decomposition which solves mixed integer optimal control problems miocps to epsilon optimality by solving one continuous nonlinear program and one mixed integer linear program milp within this work we analyze the integrality gap of miocps under mdt constraints by providing tight upper bounds on the milp subproblem we suggest different rounding schemes for constructing mdt feasible control solutions e g we propose a modification of sum up rounding a numerical study supplements the theoretical results and compares objective values of integer feasible and relaxed solutions

on the time discretization of control systems this paper develops an approach for obtaining discrete approximations to nonlinear affine control systems that are of higher than first order of accuracy with respect to the discretization step h the approach consists of two parts first the set cal u of measurable admissible controls is replaced by an appropriate finite dimensional subset cal u n then the differential equations corresponding to controls from cal u n which are in a reasonable sense regular are discretized by single step discretization methods the main result estimates the accuracy in the first part measured in terms of a prescribed collection of performance indexes the result can be interpreted both in the context of approximation of optimal control problems and in the context of approximation of the reachable set in the first case accuracy o h2 is proven for appropriate runge kutta type discretization methods without explicitly or implicitly requiring any regularity of the optimal solutions in the case of a convex reachable set we obtain o h2 approximation with respect to the hausdorff distance and o h1 5 accuracy in the nonconvex case an application to the time aggregation of discrete time control systems is also presented

sampling decisions in optimum experimental design in the light of pontryagin s maximum principle optimum experimental design oed problems are optimization problems in which an experimental setting and decisions on when to measure the so called sampling design are to be determined such that a follow up parameter estimation yields accurate results for model parameters in this paper we use the interpretation of oed as optimal control problems with a very particular structure for the analysis of optimal sampling decisions we introduce the information gain function motivated by an analysis of necessary conditions of optimality we highlight differences between problem formulations and propose to use a linear penalization of sampling decisions to overcome the intrinsic ill conditioning of oed the results of this paper are independent from the actual numerical method to compute the solution to the oed problem and of the question of local and global optima

on infinite order and fully nonlinear partial differential evolution equations abstract in this paper we study the existence uniqueness and propagation of regularity to infinite order partial differential evolution equations our approach is essentially functional and brings interesting results even when we restrict ourselves to finite order equations

transition time optimization for switched mode dynamical systems this note considers the problem of determining optimal switching times at which mode transitions should occur in multimodal hybrid systems it derives a simple formula for the gradient of the cost functional with respect to the switching times and uses it in a gradient descent algorithm much of the analysis is carried out in the setting of optimization problems involving fixed switching mode sequences but a possible extension is pointed out for the case where the switching mode sequence is a part of the variable numerical examples testify to the viability of the proposed approach

exact wkb analysis of a schrodinger equation with a merging triplet of two simple poles and one simple turning point ii its relevance to the mathieu equation and the legendre equation abstract we develop the exact wkb analysis of an m2p1t merging two simple poles and one simple turning point schrodinger equation in part ii using a wkb theoretic transformation to the algebraic mathieu equation constructed in part i we calculate the alien derivative of its borel transformed wkb solutions at each fixed singular point relevant to the simple poles through the analysis of borel transformed wkb solutions of the legendre equations in the course of the calculation of the alien derivative we make full use of microdifferential operators whose symbols are given by the infinite series that appear in the coefficients of the algebraic mathieu equation and the legendre equation

on the exact wkb analysis of higher order simple pole type operators abstract higher order simple pole type operators that is higher order linear ordinary differential operators with a large parameter whose coefficients have simple poles at the origin are discussed from the viewpoint of the exact wkb analysis making use of the technique of microdifferential operators we clarify the singularity structure of the borel transform of their wkb solutions

direct methods with maximal lower bound for mixed integer optimal control problems many practical optimal control problems include discrete decisions these may be either time independent parameters or time dependent control functions as gears or valves that can only take discrete values at any given time while great progress has been achieved in the solution of optimization problems involving integer variables in particular mixed integer linear programs as well as in continuous optimal control problems the combination of the two is yet an open field of research we consider the question of lower bounds that can be obtained by a relaxation of the integer requirements for general nonlinear mixed integer programs such lower bounds typically suffer from a huge integer gap we convexify with respect to binary controls and relax the original problem and prove that the optimal solution of this continuous control problem yields the best lower bound for the nonlinear integer problem building on this theoretical result we present a novel algorithm to solve mixed integer optimal control problems with a focus on discrete valued control functions our algorithm is based on the direct multiple shooting method an adaptive refinement of the underlying control discretization grid and tailored heuristic integer methods its applicability is shown by a challenging application the energy optimal control of a subway train with discrete gears and velocity limits

exact wkb and abelianization for the t_3 equation we describe the exact wkb method from the point of view of abelianization both for schrodinger operators and for their higher order analogues opers the main new example which we consider is the t 3 equation an order 3 equation on the thrice punctured sphere with regular singularities at the punctures in this case the exact wkb analysis leads to consideration of a new sort of darboux coordinate system on a moduli space of flat mathrm sl 3 connections we give the simplest example of such a coordinate system and verify numerically that in these coordinates the monodromy of the t 3 equation has the expected asymptotic properties we also briefly revisit the schrodinger equation with cubic potential and the mathieu equation from the point of view of abelianization

exact capacity expressions for dual branch ricean mimo channels in this paper we derive exact expressions for the mutual information mi probability density function and the mi cumulative distribution function of dual branch multipleinput multiple output mimo systems either two transmit tx or two receive rx antennas communicating over a ricean channel with no constraint on the rank of the line of sight los channel matrix this is done both for the case where the channel is only known at the rx and for the case where the channel is known both at the tx and the rx as an example to evaluate the expressions we employ uniform linear arrays and give expressions for the eigenvalues of the los channel matrix for the dual branch case

capacity of multiple antenna systems with both receiver and transmitter channel state information the capacity of multiple antenna systems operating in rayleigh flat fading is considered under the assumptions that channel state information csi is available at both transmitter and receiver and that the transmitter is subjected to an average power constraint first the capacity of such systems is derived for the special case of multiple transmit antennas and a single receive antenna the optimal power allocation scheme for such a system is shown to be a water filling algorithm and the corresponding capacity is seen to be the same as that of a system having multiple receive antennas with a single transmitter antenna whose outputs are combined via maximal ratio combining a suboptimal adaptive transmission technique that transmits only over the antenna having the best channel is also proposed for this special case it is shown that the capacity of such a system under the proposed suboptimal adaptive transmission scheme is the same as the capacity of a system having multiple receiver antennas with a single transmitter antenna combined via selection combining next the capacity of a general system of multiple transmitter and receiver antennas is derived together with an equation that determines the cutoff value for such a system the optimal power allocation scheme for such a multiple antenna system is given by a matrix water filling algorithm in order to eliminate the need for cumbersome numerical techniques in solving the cutoff equation approximate expressions for the cutoff transmission value are also provided it is shown that compared to the case in which there is only receiver csi large capacity gains are available with optimal power and rate adaptation schemes the increased capacity is shown to come at the price of channel outage and bounds are derived for this outage probability

practical matrix completion and corruption recovery using proximal alternating robust subspace minimization low rank matrix completion is a problem of immense practical importance recent works on the subject often use nuclear norm as a convex surrogate of the rank function despite its solid theoretical foundation the convex version of the problem often fails to work satisfactorily in real life applications real data often suffer from very few observations with support not meeting the randomness requirements ubiquitous presence of noise and potentially gross corruptions sometimes with these simultaneously occurring this paper proposes a proximal alternating robust subspace minimization method to tackle the three problems the proximal alternating scheme explicitly exploits the rank constraint on the completed matrix and uses the ell 0 l 0 pseudo norm directly in the corruption recovery step we show that the proposed method for the non convex and non smooth model converges to a stationary point although it is not guaranteed to find the global optimal solution in practice we find that our algorithm can typically arrive at a good local minimizer when it is supplied with a reasonably good starting point based on convex optimization extensive experiments with challenging synthetic and real data demonstrate that our algorithm succeeds in a much larger range of practical problems where convex optimization fails and it also outperforms various state of the art algorithms

quantization of complex lagrangian submanifolds let be a smooth lagrangian submanifold of a complex symplectic manifold x we construct twisted simple holonomic modules along in the stack of deformationquantization modules on x

koopman operator based model reduction for switched system control of pdes we present a new framework for optimal and feedback control of pdes using koopman operator based reduced order models k roms the koopman operator is a linear but infinite dimensional operator which describes the dynamics of observables a numerical approximation of the koopman operator therefore yields a linear system for the observation of an autonomous dynamical system in our approach by introducing a finite number of constant controls the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem this allows us to replace each of these systems by a k rom which can be solved orders of magnitude faster by this approach a nonlinear infinite dimensional control problem is transformed into a low dimensional linear problem using a recent convergence result for the numerical approximation via extended dynamic mode decomposition edmd we show that the value of the k rom based objective function converges in measure to the value of the full objective function to illustrate the results we consider the 1d burgers equation and the 2d navier stokes equations the numerical experiments show remarkable performance concerning both solution times and accuracy

a counterexample to a conjecture of gurvits on switched systems we consider products of matrix exponentials under the assumption that the matrices span a nilpotent lie algebra in 1995 gurvits conjectured that nilpotency implies that these products are in some sense simple more precisely there exists a uniform bound l such that any product can be represented as a product of no more than i matrix exponentials this conjecture has important applications in the analysis of linear switched systems as it is closely related to the problem of reachability using a uniformly bounded number of switches it is also closely related to the concept of nice reachability for bilinear control systems the conjecture is trivially true for the case of first order nilpotency gurvits proved the conjecture for the case of second order nilpotency using the baker campbell hausdorff formula we show that the conjecture is false for the third order nilpotent case using an explicit counterexample yet the underlying philosophy behind gurvits conjecture is valid in the case of third order nilpotency namely such systems do satisfy the following nice reachability property any point in the reachable set can be reached using a piecewise constant control with no more than four switches we show that even this form of finite reachability is no longer true for the case of fifth order nilpotency

compactness and convergence rates in the combinatorial integral approximation decomposition the combinatorial integral approximation decomposition splits the optimization of a discrete valued control into two steps solving a continuous relaxation of the discrete control problem and computing a discrete valued approximation of the relaxed control different algorithms exist for the second step to construct piecewise constant discrete valued approximants that are defined on given decompositions of the domain it is known that the resulting discrete controls can be constructed such that they converge to a relaxed control in the hbox weak topology of l infty if the grid constant of this decomposition is driven to zero we exploit this insight to formulate a general approximation result for optimization problems which feature discrete and distributed optimization variables and which are governed by a compact control to state operator we analyze the topology induced by the grid refinements and prove convergence rates of the control vectors for two problem classes we use a reconstruction problem from signal processing to demonstrate both the applicability of the method outside the scope of differential equations the predominant case in the literature and the effectiveness of the approach

mixed integer models for the stationary case of gas network optimization a gas network basically consists of a set of compressors and valves that are connected by pipes the problem of gas network optimization deals with the question of how to optimize the flow of the gas and to use the compressors cost efficiently such that all demands of the gas network are satisfied this problem leads to a complex mixed integer nonlinear optimization problem we describe techniques for a piece wise linear approximation of the nonlinearities in this model resulting in a large mixed integer linear program we study sub polyhedra linking these piece wise linear approximations and show that the number of vertices is computationally tractable yielding exact separation algorithms suitable branching strategies complementing the separation algorithms are also presented our computational results demonstrate the success of this approach

on the discretization of switched linear systems abstract a bilinear single control system which can be viewed as a control formulation of a linear switched system is considered the control is restricted to take values either i in 0 1 switched system or ii in 0 1 relaxed system in more practical considerations the control is often allowed to change only at the points of a given time net with a step length h the paper investigates what is the approximation error in terms of the reachable set in the two cases i and ii the error estimates that follow directly from known results are of order h and h respectively in the present paper estimations of order h and h 1 5 are proved in a constructive way the second one makes use of the effect of non accumulation of errors established earlier by the second author

new stokes line in wkb theory the wkb theory for differential equations of arbitrary order or integral equations in one dimension is investigated the rules previously stated for the construction of stokes lines for nth order differential equations n 3 or integral equations are found to be incomplete because these rules lead to asymptotic forms of the solutions that depend on path this paradox is resolved by the demonstration that new stokes lines can arise when previously defined stokes lines cross a new formulation of the wkb problem is given to justify the new stokes lines with the new stokes lines the asymptotic forms can be shown to be independent of path in addition the wkb eigenvalue problem is formulated and the global dispersion relation is shown to be a functional of loop integrals of the action

the integer approximation error in mixed integer optimal control we extend recent work on nonlinear optimal control problems with integer restrictions on some of the control functions mixed integer optimal control problems miocp we improve a theorem sager et al in math program 118 1 109 149 2009 that states that the solution of a relaxed and convexified problem can be approximated with arbitrary precision by a solution fulfilling the integer requirements unlike in previous publications the new proof avoids the usage of the krein milman theorem which is undesirable as it only states the existence of a solution that may switch infinitely often we present a constructive way to obtain an integer solution with a guaranteed bound on the performance loss in polynomial time we prove that this bound depends linearly on the control discretization grid a numerical benchmark example illustrates the procedure as a byproduct we obtain an estimate of the hausdorff distance between reachable sets we improve the approximation order to linear grid size h instead of the previously known result with order sqrt h hackl in reachable sets control sets and their computation augsburger mathematisch naturwissenschaftliche schriften dr bernd wisner augsburg 1996 we are able to include a special ordered set condition which will allow for a transfer of the results to a more general multi dimensional and nonlinear case compared to the theorems in pietrus and veliov in syst control lett 58 395 399 2009

ergodic capacity upper bound for dual mimo ricean systems simplified derivation and asymptotic tightness an analytical upper bound on the ergodic capacity of multiple input multiple output mimo systems is deduced with the aid of a simplified approach that relies on a fundamental power normalization given their high practical usability we are particularly interested in dual configurations where both ends deploy two antenna elements contrary to the majority of related studies where only the common case of rayleigh fading is considered our analysis is extended to account for ricean fading where a deterministic line of sight los component exists in the communication link and both ends are affected by spatial correlation in the following it is shown that the proposed bound is applicable for any arbitrary signal to noise ratio snr and rank of the mean channel matrix in fact we consider both conventional and optimized mimo configurations with equal los eigenvalues moreover the tightness of the bound is explored where it is demonstrated that as the snr tends to zero the bound becomes asymptotically tight at high snrs the offset between empirical capacity and the bound is analytically computed which implies that an explicit asymptotic capacity expression can ultimately be obtained

mixed integer optimal control problems with switching costs a shortest path approach we investigate an extension of mixed integer optimal control problems by adding switching costs which enables the penalization of chattering and extends current modeling capabilities the decomposition approach consisting of solving a partial outer convexification to obtain a relaxed solution and using rounding schemes to obtain a discrete valued control can still be applied but the rounding turns out to be difficult in the presence of switching costs or switching constraints as the underlying problem is an integer program we therefore reformulate the rounding problem into a shortest path problem on a parameterized family of directed acyclic graphs dags solving the shortest path problem then allows to minimize switching costs and still maintain approximability with respect to the tunable dag parameter theta we provide a proof of a runtime bound on equidistant rounding grids where the bound is linear in time discretization granularity and polynomial in theta the efficacy of our approach is demonstrated by a comparison with an integer programming approach on a benchmark problem

third order nilpotency nice reachability and asymptotic stability we consider an affine control system whose vector fields span a third order nilpotent lie algebra we show that the reachable set at time t using measurable controls is equivalent to the reachable set at time t using piecewise constant controls with no more than four switches the bound on the number of switches is uniform over any final time t as a corollary we derive a new sufficient condition for stability of nonlinear switched systems under arbitrary switching this provides a partial solution to an open problem posed in d liberzon lie algebras and stability of switched nonlinear systems in v blondel a megretski eds unsolved problems in mathematical systems and control theory princeton univ press 2004 pp 203 207

on the exact wkb analysis of operators admitting infinitely many phases abstract to analyze differential operators whose wkb solutions admit infinitely many phases we introduce a class of differential operators of wkb type and analyze their exact wkb theoretic structure near their turning points our analysis makes full use of techniques and ideas in microlocal analysis we use a quantized contact transformation to construct a wkb solution of a differential equation of wkb type and we use a spath type division theorem for a differential operator of wkb type to study its structure near turning points as an application we show a connection formula for wkb solutions near a simple turning point