Gerhard Jank

• Prof. Dr.
• RWTH Aachen University

In this work, we derive necessary and sufficient conditions for the existence of an hierarchic equilibrium of a disturbed two player linear quadratic game with open loop information structure. A convexity condition guarantees the existence of a unique Stackelberg equilibria; this solution is first obtained in terms of a pair of symmetric Riccati eq...

In view to the decentralised problem of gas network optimisation, we model the problem as differential game where the players are the network controllable elements that communicate through nearest-neighbour network components. The controllable elements are sources and compressors. But since these do not have the same relevance within the network, i...

In this paper, the gas dynamics within the pipelines is written as a wave repetitive process, and modified in a way that the dynamics is driven by the boundary conditions. We study controllability of the system through boundary control and every agent, as well as observability of the system being steered by initial and boundary data. Next, we obtai...

In this paper we design a model based method to locate a leakage and estimate its size in a gas network, using a linearised version of an hyperbolic PDE. To do this, the problem is reduced to two identical ODEs, allowing in this way for a representation of the pressure as well as the mass flow in terms of its system of fundamental solutions. Then u...

In this paper, the gas dynamics within the pipelines is written as a wave repetitive process, and modify it in a way that the dynamics is influenced by p decision makers, namely the boundary conditions. We obtain sufficient criteria for the existence of boundary equilibrium controls as well as controllability of the different agents and observabili...

We present two different methods to obtain global existence results for solutions of nonsymmetric Riccati matrix differential equations. In the first approach we derive sufficient conditions ensuring that the spectral norm of the solutions remains uniformly bounded in an interval (-∞;to]; in a second part we make use of the linearizability of the R...

We examine disturbed linear quadratic differential games, where each player chooses his strategy according to the definition of a Nash/worst-case equilibrium. We derive sufficient conditions for the equilibrium strategies and we also give formulae for the optimal controls using solutions of Riccati differential equations. In a special case, where t...

In this paper, the gas dynamics within the pipelines is modelled as a repetitive process with smoothing. Controllability and observability criteria when the system is steered through initial and boundary data, which is achieved by an adequate choice of the homogeneity, are obtained. From the point of view of the technical applications, it seems to...

In this paper, we proffer an explicit representation of solutions for a specific class of linear repetitive processes with smoothing. This representation is used to obtain direct controllability and observability criteria of this same class of discrete time 2-D systems. Not only classical controllability properties are considered, where control of...

An indirect downsampling approach for continuous-time input/output system identification is proposed. This modus operandi was introduced to system identification through a subspace algorithm, where the input/output data set is partitioned into lower rate m subsets. Then, a state-space discrete-time model is identified by fusing the data subsets int...

This article the gas network is modelled as a repetitive process with smoothing and an explicit representation for its solution is presented. By an adequate choice of the inhomogeneity, the system is steered through boundary data control. We state explicit conditions for boundary control of the gas networks as well as observability criteria. Using...

This article presents necessary and sufficient results for existence and uniqueness of an equilibrium of a N-player disturbed Nash game with quadratic performance criteria and an affine repetitive process with smoothing describing the two dimensional 2D-system dynamics, under open loop information pattern. The gas dynamics in a single pipeline is m...

A new approach to gas leakage detection in high pressure distribution networks is proposed, where two leakage detectors are modelled as a Linear Parameter Varying (LPV) system whose scheduling signals are, respectively, intake and offtake pressures. Running the two detectors simultaneously allows for leakage location. First, the pipeline is identif...

This article presents a new indirect identification method for continuous-time systems able to resolve the problem of fast sampling. To do this, a Subspace IDentification Down-Sampling (SIDDS) approach that takes into consideration the intermediate sampling instants of the input signal is proposed. This is done by partitioning the data set into m s...

In this note, we investigate the solution of a disturbed quadratic open loop (OL) Nash game, whose underlying system is an affine differential equation and with a finite time horizon. We derive necessary and sufficient conditions for the existence/uniqueness of the Nash/worst-case equilibrium. The solution is obtained either via solving initial/ter...

In this paper a new approach to gas leakage detection in high pressure natural gas transportation networks is proposed. The pipeline is modelled as a Linear Parameter Varying (LPV) System driven by the source node massflow with the gas inventory variation in the pipe (linepack variation, proportional to the pressure variation) as the scheduling par...

In this paper a lumped transfer function (TF) model is derived for High Pressure Natural Gas Pipelines. Departing from a nonlinear partial differential equation (PDE) model a high order continuous state space (SS) linear model is obtained using a finite difference method. An infinite order TF is calculated from the SS representation and finally is...

A new approach to gas leakage detection in high pressure distribution networks is proposed, where the pipeline is modelled as a Linear Parameter Varying (LPV) System driven by the source node mass flow with the pressure as the scheduling parameter, and the system output as the mass flow at the offtake. Using a recently proposed successive approxima...

In this report a lumped transfer function model for High Pressure Natural Gas Pipelines is derived. Starting with a partial nonlinear differential equation (PDE) model a high order continuous state space (SS) linear model is obtained using a finite difference method. Next, from the SS representation an infinite order transfer function (TF) model is...

This article presents necessary and sufficient results for existence and uniqueness of an equilibrium of a N -player Nash game with quadratic performance criteria and an affine repetitive process with smoothing describing the two dimensional (2D) system dynamics, under open loop information pattern. The gas dynamics in a single pipeline is modelled...

We perform a qualitative analysis of a differential equation that was originally introduced by Stortelder, Hemker and Hemker to model the formation of thrombin, and discuss issues of controllability and stabilizability. Results include a general proof of convergence to equilibrium, and of local exponential stabilizability.

In this note, we investigate the solution of a disturbed quadratic open loop Nash game, whose underlying system is given by an affine differential equation and with a finite time horizon. We derive necessary and sufficient conditions for the existence/uniqueness of the Nash/worst-case equilibrium. The solution is obtained either via solving initial...

In this paper, we propose to model the optimisation of gas networks as a disturbed LQ game, where each player chooses his strategy according to a modified Nash condition under OL information structure. The dynamics in the network, as well as the coupling conditions, are modelled through a DAE system, and as a consequence a network with loops and no...

In this paper we consider some classes of discrete- continuous 2-D models in order to build a finite dimensional linear state space model of a gas distribution network. The models introduced are shown to be suitable for handling problems of optimal control of pressure and flow in gas transport units. The focus is on the development of a comprehensi...

The plasma coagulation system is a biochemical chain reaction where inactive proenzymes are converted to active enzymes in a cascade pattern. One of the problems encountered in the modelling of thrombin generation in plasma is that neither the reaction mechanism nor the reaction constants and initial concentrations are precisely known. Therefore, t...

We consider disturbed linear 2D-systems of Fornasini–Marchesini type in the continuous time case. These systems are also named Goursat-type systems. Conditions for unique solvability of the disturbed optimal control problem with a quadratic cost functional are obtained. The disturbed or worst case optimal control guarantees to minimize the cost fun...

In this paper two-player Nash differential games, on an infinite time horizon, with two different information structures have been considered: the open loop and the deterministic feedback information structure. The performance indices were assumed to be of quadratic type and the constraint to be a linear positive differential system. As the main re...

This paper considers some classes of discrete 2D models to construct the discrete time, finite dimensional linear state space model of a gas distribution network. The models introduced are used to present some methods of optimal control applicable to problem of pressure and flow control in output nodes of gas transport networks. The focus is on the...

Repetitive processes are a distinct class of two-dimensional systems (i.e. information propagation in two independent directions) of both systems theoretic and applications interest. They cannot be controlled by direct extension of existing techniques from either standard (termed 1D here) or two-dimensional (2D) systems theory. In this paper we def...

In this note we study a fixed point iteration approach to solve algebraic Riccati equations as they appear in general two player Nash differential games on an infinite time horizon, where the information structure is of open loop type. We obtain conditions for existence and uniqueness of non-negative solutions. The performance of the numerical algo...

Repetitive processes are a distinct class of two-dimensional systems (i.e. information propagation in two independent directions) of both systems theoretic and applications interest. They cannot be controlled by direct extension of existing techniques from either standard (termed 1D here) or two-dimensional (2D) systems theory. in this paper we def...

In this paper we define positive games, which are based on the notion of positive systems. Especially, two player open loop Nash games on the infinite time horizon are studied. It is known (see e.g. [11] or [12]), that the exis-tence of a unique Nash equilibrium is mainly related with the existence of a left-right stabilizing solution to the algebr...

We consider disturbed linear 2D-systems of Fornasini-Marchesini type in the continuous time case. These systems are also named Goursat-type systems. Conditions for unique solv-ability of the disturbed optimal control problem with a quadratic cost functional are obtained. The disturbed or worst case optimal control guarantees to minimize the cost fu...

We examine disturbed linear-quadratic games, where each player chooses his strategy ac- cording to a modified Nash equilibrium model under open-loop information structure. We give conditions for the existence and uniqueness of such an equilibrium. We also show how these conditions are related to certain Riccati difference equations and a boundary v...

This chapter investigates linear quadratic open loop Nash games on the infinite time horizon. The main theorem states a necessary and sufficient condition for the existence of a unique Nash equilibrium. It is obtained under the weakest conditions possible on the coefficients of the game. In the case considered, this theory relates the invertibility...

In this note we study a fixpoint iteration approach to solve algebraic Riccati equations as they appear in general two player Nash differential games on an infinite time horizon, where the information structure is of open loop type. We obtain conditions for existence and uniqueness of non negative solutions. The performance of the numerical algorit...

In this paper we deal with Nash equilibrium in linear - qudratic differential games when a discount rate is introduced in the players utility function. It is shown that coupled Riccati equations occurring in such a situation may be considered as special higher order non-symmetric single Riccati equation. Then, some results are adapted to the case o...

The existence and uniqueness of open-loop Nash equilibria in linear-quadratic discrete time games are studied. The approach used is the construction of a value function which leads to existence assertions in terms of solvability of certain symmetric and nonsymmetric Riccati difference equations. Under the condition of solvability of the Riccati equ...

Open-loop Stackelberg games are conceptual very interesting particularly for applications which contain a hierarchical struc- ture. The existence of unique Stackelberg equilibria was shown to be tied to the existence of solutions to certain nonsymmetric Riccati equations, which are hard to solve. The paper reveals a connection between solutions of...

In Chapter 5 we assume that A, N, Q = Q*, S = S*: ℝ → ℂnxn, B: ℝ → ℂnxm, b : ℝ → ℂn, M : ℝ → ℂmxm, R : ℝ → ℂmxn and C : ℝ → ℂpxn are piecewise continuous, locally integrable, ω-periodic functions and we consider the periodic Hermitian Riccati differential equation $$ \dot X = - A^* (t)X - XA(t) - Q(t) + XS(t)X, $$ (PHRDE) corresponding to the w-per...

The purpose of this chapter is to study coupled, non-symmetric and generalized Riccati equations occurring in different control problems. Three main topics are addressed: Dynamic games, linear systems with Markovian jumps and generalized Riccati equations appearing in stochastic control. In all these cases when quadratic criteria have to be minimiz...

This chapter is dedicated to the theory of Hermitian Riccati differential equations (HRDE), which are of importance in various fields of applications, as e.g., the linear quadratic optimal problem, differential games, differential geometry, fac­torization problems and spectral theory. Since the linear differential system that, according to Radon’s...

The main subject of this book is matrix Riccati differential equations; by definition, in this book, these are differential equations which can be written in the form $$ \dot W = M_{21} (t) + M_{22} (t)W - WM_{11} (t) - WM_{12} (t)W,t \in \mathcal{I}, $$ (RDE) with W, M21(t) ∈ ℂmxn, M22(t) ∈ ℂmxm, M11(t) ∈ ℂnxn, M12(t) ∈ ℂnxm for t ∈ \( \mathcal{I}...

In this chapter we consider two applications of the Riccati equation theory of indefinite sign to some typical problems in robust control: the Nehari problem and the disturbance attenuation problem. Historically, the first solution to the disturbance attenuation problem was given by reducing it to a four block Nehari problem. However, we will follo...

Riccati differential equations are among the simplest non-linear differential equa­tions and, clearly, then initial value problems can be solved locally. But, in con­trast to linear systems of differential equations, their solutions may show the phe­nomenon of finite escape time . This generally means that after a finite time interval the solution...

In this chapter we consider symmetric differential Riccati equations with real, continuous and bounded matrix-valued coefficients. We investigate the existence of (global) stabilizing real solutions and extend the theory to the case of quadratic indefinite sign terms. This extension is formulated in the framework of the generalized Popov—Yakubovich...

In this section we recall some well-known facts about first order linear systems of differential equations with piecewise continuous, locally bounded coefficients and, in the case of constant coefficients, about the corresponding algebraic equations.

The terminology ‘Riccati theory’ is inspired by the book [I0W99] of Ionescu, Oará and Weiss. Therein, Riccati theory stands for the linkage between existence results for stabilizing solutions to the symmetric algebraic Riccati equation, the invertibility of the Toeplitz operator associated with the input-output operator of the underlying Hamiltonia...

We examine disturbed linear quadratic differential games, where each player chooses his strategy according a modified Nash equilibrium model under open-loop information structure. We give conditions for the existence and uniqueness of such an equilibrium. We also show how these conditions are related to certain Riccati differential equations. We co...

We examine disturbed linear-quadratic games, where each player chooses his strategy according to a modified Nash equilibrium model under open-loop information structure. We give conditions for the existence and uniqueness of such an equilibrium. We also show how these conditions are related to certain Riccati di#erence equations and a boundary valu...

We consider linear 2-D systems of Fornasini-Marchesini type in the continuous-time case with non-constant coefficients. Using an explicit representation of the solutions by utilizing the Riemann-kernel of the equation under consideration, we obtain controllability and observability criteria in the case of the inhomogeneous equation, where control i...

We present existence and uniqueness results for a hierarchical or Stackelberg equilibrium in a two-player differential game with open-loop information structure. There is a known convexity condition ensuring the existence of a Stackelberg equilibrium, which was derived by Simaan and Cruz (Ref. 1). This condition applies to games with a rather nonco...

This paper investigates linear quadratic open loop Nash games on the infinite time horizon. The main theorem states a necessary and sufficient condition for the existence of a unique Nash equilibrium. It is obtained under the weakest conditions possible on the coefficients of the game. Especially, the convexity condition is relaxed. The starting po...

This paper studies the condition under which a geometric characterization of instant local and global controllability of control constrained continuous 2-D systems can be given. Necessary and sufficient conditions to get instant controllability are derived. The importance of the imposed assumption is supported by counterexamples. The results enligh...

We present several methods to obtain global existence results for solutions of non-symmetric Riccati matrix differential equations and for generalized or perturbed symmetric Riccati differential equations. One approach is to derive sufficient conditions ensuring that the spectral norm of the solutions remain uniformly bounded in an interval (−∞, t0...

We study the asymptotic behavior of difference equations appearing in the necessary opti-mality conditions of noncooperative open loop Nash and Stackelberg games. Since these equations are coupled nonsymmetric Riccati difference equations, their qualitative behavior is essentially different from that of standard (symmetric) Riccati equations. Moreo...

We present an algorithm for the solution of a nontrivial coupled system of algebraic Riccati equations appearing in risk sensitive control problems. Moreover, we use comparison methods to derive non-blowup conditions for the solutions of a corresponding terminal value problem for coupled systems of Riccati differential equations

The authors study the solution of generalized matrix Riccati differential equations as they appear in differential games theory with various information structures. A numerical approximation is proposed for the generalized Riccati differential equation arising from an optimally controlled differential game with memoryless feedback. This method make...

We are investigating the dependence of the phase portrait and in particular ofthe constant solutions ofgame (or H∞-type) Riccati differential equations (RDEμ,) on the parameter μ E R. Moreover we give sufficient conditions for the existence of the solution Wμ, of (RDEμ,) with Wμ(tf) = Wf for t ≤ tf and also for the existence of at least one pole of...

We present comparison and global existence theorems for solutions of generalized matrix Riccati differential and difference equations. Moreover we obtain existence and comparison results for the maximal solutions of the corresponding generalized algebraic Riccati equations. For the symplectic matrix Riccati differential equation we derive sufficien...

In this paper we investigate generalized Riccati differential and difference equations obtained from standard Riccati equations by adding a semidefinite perturbation term. For such equations we give results on the monotonic dependence of the solutions on the coefficients and initial values as well as results on convergence of solutions.

We describe the iterative behavior of Moebius transformations with values in the quaternions. A classification of the main conjugacy classes is given. The dynamics turns out to be similar to the complex case, with the exception of the case of a two dimensional fixed point manifold. For Moebius transformations mapping the unit ball onto itself an an...

Presents comparison and global existence results for solutions of coupled matrix Riccati differential equations appearing in closed loop Nash games and in mixed H₂/H_∞-type problems. Convergence of solutions is established for the diagonal case, solutions of the corresponding algebraic equations are discussed using numerical ex...

Necessary conditions for the Newton polygon associated with a general differential equation algebraic in the unknown function and its Schwarzian derivative are presented in the case of existence of an admissible solution (in Nevanlinna sense). Moreover, an admissible solution is shown to satisfy a stronger version of Nevanlinna's defect relation.

The problem of computing optimal production rates for a multimachine multiproduct manufacturing system subject to machine failure is considered. The inventory balance equation is represented by a discrete-time flow model with Markovian jumps to take into account machine breakdown. When defining quadratic cost functions, the associated optimal contr...

A necessary and sufficient condition for the existence of a positive-semidefinite solution of the coupled algebraic discrete-time Riccati-like equation occurring in Markovian jump control problems is derived. By verifying a simple matrix inequality, it is shown that such a solution exists and can be obtained as a limit of a monotonic sequence. This...

We prove a fundamental representation formula for all solutions of the matrix Riccati dierential equation ( RDE) and of the corresponding algebraic Riccati equation (ARE). This formula contains the complete information on the phase portrait of (RDE) and on the structure of the set of all solutions of ( ARE). In particular we describe all con- stant...

A new necessary and sufficient condition for the existence of a positive semidefinite solution of coupled Riccati equations occurring in jump linear systems is derived. By verifying a Riccati inequality it is shown that such a solution exists; in addition two numerical algorithms are given to compute it. An example is given to illustrate the propos...

The asymptotic behaviour of solutions as t → ∞ for coupled matrix Riccati equations occurring in open-loop linear-quadratic Nash games is studied in this paper. A general formula representing all possible solutions is given. Necessary conditions for constant real solutions are derived and an estimate for the rate of convergence is obtained. Two exa...

Theoretical and experimental investigations were performed with cold water models of bottom-blown metallurgical ladles in order to develop a mathematical description of the mixing processes in such reactors. This analytical model is based on the subdivision of the reactor into interacting subspaces, whereby each subspace is regarded as an ideal mix...

In the present paper, we make use of the method of asymptotic integration to get estimates on those regions in the complex plane where singularities and critical points of solutions of the Matrix-Riccati differential equation with polynomial co-efficients may appear. The result is that most of these points lie around a finite number of permanent cr...

we prove the following theorem for characteristic functions, which is an extension of theorems of Marcinkiew and Luckas: let P be a polynomial of degree m and let exp i(z) denote the l-times iteratd exponential function. Assume futher g(z) is an entire function with the property (here M(r,g) denotes the usual maximum modulus of the entire function...

In this paper we use Bergman's Kernel Method (BKM) for solving conformal mapping problems. In contrast to most conformal mapping techniques, the approximation of the solution is an analytic function. The main drawback of the method, however, is that it involves an orthonormalization process and thus is numerically unstable. This disadvantage can be...

Without Abstract

Über diese Arbeit wurde auf einem “Tag der Funktionentheorie” am 15. und 16.6.1984 an der RWTH Aachen vom ersten Autor berichtet. This lecture is concerned with the asymptotic behavior of solutions of linear differential equations and iinear systems in the irregular singular case. In contrast to the existing theory of asymptotic integration (compar...

Über diese Arbeit wurde auf einem “Tag der Funktionenthcorie” am 15. und 16.6.1984 an der RWTH Aachen vom zweiten Verfasser berichtet. Recently E. Mues and N. Steinmetz and independently G. G. Gundersen proved that if a nonconstant meromorphic function f shares two different non-zero values a and b (counting multiplicities) with its first derivativ...

Let w be a meromorphic function and where aλ,bλ,αλ, Pλ denote polynomials g an entire function and Q. Q1rational functions.In this paper we investigate the growth of meromorphic solutions of the following equations. where H,H1,H2 are of the form (1.1) and P is a polynomial andwhere Q1 satisfy the condition

Als wichtige Vorbereitung für die nächsten drei Abschnitte wollen wir die Poisson-Jensen-Nevanlinnasche Formel herleiten. Ausgangspunkt der Betrachtungen ist eine der Cauchyschen Integralformel verwandte Darstellung für holomorphe Funktionen bzw. deren Realteile.

Eine in der ganzen komplexen Ebene C holomorphe Funktion g heißt ganze Funktion. Eine solche Funktion besitzt bekanntlich eine in der ganzen Ebene konvergente Taylorreihenentwicklung \(g\left( z \right) = \sum\limits_{n = 0}^\infty {{a_n}{z^n}}\). Hat diese Reihe nur end-lich viele Glieder, so nennt man g Polynom oder ganz rational, hat sie unendli...

Bei der konformen Abbildung von Polygongebieten (man vgl. z. B. Ahlfors [1], S. 236) erhält man mit der Schwarz-Christoffel-Formel in vielen Fällen Differentialgleichungen der Form$${w'^n} = {a_0} + {a_1}w + ... + {a_k}{w^k},\quad k,n \in IN,\quad {a_j} \in \mathbb{C},\quad j = 0,1,...,k

Eine ganze Funktiong ohne Nullstellen besitzt, wie wir beim Beweis von Satz 2.1 gezeigt haben, eine Darstellung der Form$$g\left( z \right) = {e^{h\left( z \right)}}$$ (10.1) wobeih eine ganze Funktion ist.

Es sei f eine meromorphe und g eine ganze Funktion. Der Fall, daß f oder g konstant sind, sei im folgenden stets ausgeschlossen. Dann ist auch die zusammengesetzte Funktion (f∘g)(z) = f(g(z)) wieder meromorph (und nicht konstant). Ist f sogar ganz, so ist auch f∘g ganz. In diesem Kapitel beschäftigen wir uns mit folgenden Fragen: Wie lassen sich T(...

In this paper we give an elementary proof of the famous theorem of J. Malmquist from 1913 and its refinement given by N. Steinmetz 1978. This result says, that if a differential equation of type (1.1) has a transcendental meromorphic solution, then it can be transformed into one of the six normalforms given in theorem 3.2 or a power of it. In this...

Let f be a meromorphic (entire) function and denote by M(rf) the maximum modulusT(rf) Nevannlinna's characteristicn(raf) = n(ra) the number of a-points of f in the counting function. Suppose P is the Weierstraß productthen our main result is the inequalityIf x ≥ 0 we set . The l-order and the lower l-order of a meromorphic (entire) function f are d...

By the aid of the comparison function , one can define the l-order of an entire function. Using the comparison function we define just like in the classical manner (l,k)-types and get Weierstraß products P with the properties , if l ≥ 1, and if l ≥ 3 and l ≤ 0k ≤ l-2, where σl,k(0) denotes the (lk)-type of the zeros of the product.