On the calculation of neutral subspaces of a matrix

A technique for constructing solutions to the quadratic matrix equation X ^T DX +AX + X ^T B + C = 0 is outlined. It is similar to the well-known Schur approach for solving algebraic Riccati equations.     

Computation of the stabilizing solution of game theoretic Riccati equation arising in stochastic <Emphasis Type="Italic">H</Emphasis><Subscript> ∞ </Subscript> control problems

In this paper, the problem of the numerical computation of the stabilizing solution of the game theoretic algebraic Riccati equation is investigated. The Riccati equation under consideration occurs in connection with the solution of the H _∞ control problem for a class of stochastic systems affected by state dependent and control dependent white noise. The stabilizing solution of the considered game theoretic Riccati equation is obtained as a limit of a sequence of approximations constructed based on stabilizing solutions of a sequence of algebraic Riccati equations of stochastic control with definite sign of the quadratic part. The efficiency of the proposed algorithm is demonstrated by several numerical experiments.     

A noniterative algebraic solution for Riccati equations satisfying two-point boundary-value problems

A noniterative algebraic method is presented for solving differential Riccati equations which satisfy two-point boundary-value problems. This class of numerical problems arises in quadratic optimization problems where the cost functionals are composed of both continuous and discrete state penalties, leading to piecewise periodic feedback gains. The necessary condition defining the solution for the two-point boundary value problem is cast in the form of a discrete-time algebraic Riccati equation, by using a formal representation for the solution of the differential Riccati equation. A numerical example is presented which demonstrates the validity of the approach.The authors would like to thank Dr. Fernando Incertis, IBM Madrid Scientific Center, who reviewed this paper and pointed out that the two-point boundary-value necessary condition could be manipulated into the form of a discrete-time Riccati equation. His novel approach proved to be superior to the authors' previously proposed iterative continuation method.     

On homotopy varieties

Given an algebraic theory T, a homotopy T-algebra is a simplicial set where all equations from T hold up to homotopy. All homotopy T-algebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the characterization of varieties.     

Quadratic optimal control of stable well-posed linear systems

We consider the infinite horizon quadratic cost minimization problem for a stable time-invariant well-posed linear system in the sense of Salamon and Weiss, and show that it can be reduced to a spectral factorization problem in the control space. More precisely, we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution. If both the system and the spectral factor are regular, then the feedback operator can be expressed in terms of the Riccati operator, and the Riccati operator is a positive self-adjoint solution of an algebraic Riccati equation. This Riccati equation is similar to the usual algebraic Riccati equation, but one of its coefficients varies depending on the subspace in which the equation is posed. Similar results are true for unstable systems, as we have proved elsewhere.

     

Representation of solutions of n-order Riccati equation via generalized trigonometric functions

In this work we suggest a systematic method of construction of solutions of the n-order Riccati equation with constant coefficients in a field from the set of generalized trigonometric functions. The generalized trigonometric functions satisfy the system of evolution equations generated by the companion matrix of n  -order polynomial. The set of trigonometric functions depend of (n−1)$(n-1)$ variables formally expressed by series of exponential functions. In a particular case, the second order Riccati equation with constant coefficients is isomorphic to the evolution equation generated by the companion matrix of the associated quadratic polynomial. It is shown that the n>2$n>2$ order Riccati equation with coefficients in a field is derived from a linear system of evolution equations generated by companion matrix of the associated n  -order polynomial under (n−2)$(n-2)$ constraints.     

Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems

Summary This paper considers the optimal quadratic cost problem (regulator problem) for a class of abstract differential equations with unbounded operators which, under the same unified framework, model in particular «concrete» boundary control problems for partial differential equations defined on a bounded open domain of any dimension, including: second order hyperbolic scalar equations with control in the Dirichlet or in the Neumann boundary conditions; first order hyperbolic systems with boundary control; and Euler-Bernoulli (plate) equations with (for instance) control(s) in the Dirichlet and/or Neumann boundary conditions. The observation operator in the quadratic cost functional is assumed to be non-smoothing (in particular, it may be the identity operator), a case which introduces technical difficulties due to the low regularity of the solutions. The paper studies existence and uniqueness of the resulting algebraic (operator) Riccati equation, as well as the relationship between exact controllability and the property that the Riccati operator be an isomorphism, a distinctive feature of the dynamics in question (emphatically not true for, say, parabolic boundary control problems). This isomorphism allows one to introduce a «dual» Riccati equation, corresponding to a «dual» optimal control problem. Properties between the original and the «dual» problem are also investigated.Research partially supported by the National Science Foundation under Grant NSF-DMS-8301668 and by the Air Force Office of Scientific Research under Grant AFOSR-84-0365.     

Positive semigroups and algebraic Riccati equations in Banach spaces

We generalize Wonham’s theorem on solvability of algebraic operator Riccati equations to Banach spaces, namely there is a unique stabilizing solution to \(A^*P+PA-PBB^*P+C^*C=0\) when (A, B) is exponentially stabilizable and (C, A) is exponentially detectable. The proof is based on a new approach that treats the linear part of the equation as the generator of a positive semigroup on the space of symmetric operators from a Banach space to its dual, and the quadratic part as an order concave map. A direct analog of global Newton’s iteration for concave functions is then used to approximate the solution, the approximations converge in the strong operator topology, and the convergence is monotone. The linearized equations are the well-known Lyapunov equations of the form \(A^*P+PA=-Q\), and semigroup stability criterion in terms of them is also generalized.     

Classification of quadratic systems admitting the existence of an algebraic limit cycle

In the paper we find a set of necessary conditions that must be satisfied by a quadratic system in order to have an algebraic limit cycle. We find a countable set of ?5 parameter families of quadratic systems such that every quadratic system with an algebraic limit cycle must, after a change of variables, belong to one of those families. We provide a classification of all the quadratic systems which can have an algebraic limit cycle based on geometrical properties of the embedding of the system in the Poincaré compactification of R². We propose names for all the classes we distinguish and we classify all known examples of quadratic systems with algebraic limit cycle. We also prove the integrability of certain classes of quadratic systems.     

On the growth of solutions of certain second-order nonlinear differential equations

In this paper, we determine the growth of real-valued solutions of certain second-order algebraic differential equations. Our main result, together with a result of G. Valiron, shows that if y₀ is an entire function which has only real, nonnegative coefficients in its power series around the origin, and which is a solution of a quadratic second-order algebraic differential equation, then y₀ satisfies a growth estimate of the form,y₀(x) ? exp(exp x^c), where c is a constant, for all sufficiently large x. The determination of the growth of such solutions was an open problem since the Valiron-Wiman theory fails to provide any information on growth, if the equation possesses a solution of infinite order of growth.     

Linear relations among algebraic solutions of differential equations

For any univariate polynomial with coefficients in a differential field of characteristic zero and any integer, q, there exists an associated nonzero linear ordinary differential equation (LODE) with the following two properties. Each term of the LODE lies in the differential field generated by the rational numbers and the coefficients of the polynomial, and the qth power of each root of the polynomial is a solution of this LODE. This LODE is called a qth power resolvent of the polynomial. We will show how one can get a resolvent for the logarithmic derivative of the roots of a polynomial from the αth power resolvent of the polynomial, where α is an indeterminate that takes the place of q. We will demonstrate some simple relations among the algebraic and differential equations for the roots and their logarithmic derivatives. We will also prove several theorems regarding linear relations of roots of a polynomial over constants or the coefficient field of the polynomial depending upon the (nondifferential) Galois group. Finally, we will use a differential resolvent to solve the Riccati equation.     

Exact slow-fast decomposition of the singularly perturbed matrix differential Riccati equation

In this paper the Hamiltonian matrix formulation of the Riccati equation is used to derive the reduced-order pure-slow and pure-fast matrix differential Riccati equations of singularly perturbed systems. These pure-slow and pure-fast matrix differential Riccati equations are obtained by decoupling the singularly perturbed matrix differential Riccati equation of dimension n₁+n₂ into the pure-slow regular matrix differential Riccati equation of dimension n₁ and the pure-fast stiff matrix differential Riccati equation of dimension n₂. A formula is derived that produces the solution of the original singularly perturbed matrix differential Riccati equation in terms of solutions of the pure-slow and pure-fast reduced-order matrix differential Riccati equations and solutions of two reduced-order initial value problems. In addition to its theoretical importance, the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear time-invariant systems independently in pure-slow and pure-fast time scales.     

On the Minimal Nonnegative Solution of Nonsymmetric Algebraic Riccati Equation

We study perturbation bound and structured condition number about the minimal nonnegative solution of nonsymmetric algebraic Riccati equation, obtaining a sharp perturbation bound and an accurate condition number. By using the matrix sign function method we present a new method for finding the minimal nonnegative solution of this algebraic Riccati equation. Based on this new method, we show how to compute the desired M-matrix solution of the quadratic matrix equation X^2 - EX - F = 0 by connecting it with the nonsymmetric algebraic Riccati equation, where E is a diagonal matrix and F is an M-matrix.     

On algebraic Riccati equations associated with M-matrices

We consider the algebraic Riccati equation for which the four coefficient matrices form an M-matrix K. When K is a nonsingular M-matrix or an irreducible singular M-matrix, the Riccati equation is known to have a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M-matrix. Under a regularity assumption on the M-matrix K, we show that the Riccati equation still has a minimal nonnegative solution. We also study the properties of this particular solution and explain how the solution can be found by existing methods.     

A Point Model for the Free Cyclic Submodules over Ternions

We show that the set of all (unimodular and non-unimodular) free cyclic submodules of T ², where T is the ring of ternions over a commutative field, admits a point model in terms of a smooth algebraic variety.     

A class of uniform functions and its relationship with the class of measurable functions

Borel, Lebesgue, and Hausdorff described all uniformly closed families of real-valued functions on a set T whose algebraic properties are just like those of the set of all continuous functions with respect to some open topology on T. These families turn out to be exactly the families of all functions measurable with respect to some σ-additive and multiplicative ensembles on T. The problem of describing all uniformly closed families of bounded functions whose algebraic properties are just like those of the set of all continuous bounded functions remained unsolved. In the paper, a solution of this problem is given with the help of a new class of functions that are uniform with respect to some multiplicative families of finite coverings on T. It is proved that the class of uniform functions differs from the class of measurable functions.     

Lamé equations with algebraic solutions

In this paper we study Lamé equations L_n,By=0 in so-called algebraic form, having only algebraic functions as solution. In particular we provide a complete list of all finite groups that occur as the monodromy groups, together with a list of examples of such equations. We show that the set of such Lamé equations with is countable, up to scaling of the equation. This result follows from the general statement that the set of equivalent second-order equations, having algebraic solutions and all of whose integer local exponent differences are 1, is countable.     

ON THE MINIMAL NONNEGATIVE SOLUTION OF ONSYMMETRIC ALGEBRAIC RICCATI EQUATION

We study perturbation bound and structured condition number about the minimalnonnegative solution of nonsymmetric algebraic Riccati equation,obtaining a sharp per-turbation bound and an accurate condition number.By using the matrix sign functionmethod we present a new method for finding the minimal nonnegative solution of this al-gebraic Riccati equation.Based on this new method,we show how to compute the desiredM-matrix solution of the quadratic matrix equation X~2-EX-F=0 by connecting itwith the nonsymmetric algebraic Riccati equation,where E is a diagonal matrix and F isan M-matrix.     

A systematic method for solving differential-difference equations

A systematic method for searching travelling-wave solutions to differential-difference equations (DDEs) is proposed in the paper. First of all, we introduce Bäcklund transformations for the standard Riccati equation which generate new exact solutions by using its simple and known solutions. Then we introduce a kind of formal polynomial solutions to DDEs and further determine the explicit forms by applying the balance principle. Finally, we work out exact solutions of the DDEs via substituting the form solutions and solving over-determined algebraic equations with the help of Maple. As illustrative examples, we obtain the travelling-wave solutions of the (2 + 1)-dimensional Toda lattice equation, the discrete modified KdV (mKdV) equation, respectively.     

The homogeneous projective transformation of general quadratic matrix equations

^** Email: vassilios.tsachouridis{at}ieee.org^*** Email: basil.kouvaritakis{at}eng.ox.ac.uk Algebraic quadratic equations are a special case of a singlegeneralized algebraic quadratic matrix equation (GQME). Hence,the importance of that equation in science and engineering isevident. This paper focus on the study of solutions of thatGQME and a unified framework for the characterization and identificationof solutions at infinity and of finite solutions of generalquadratic algebraic matrix equations is presented. The analysisis based on the concept of homogeneous projective transformationfor general polynomial systems (Morgan, 1986). In addition,a numerical error analysis for the computed solutions is providedfor the assessment of numerical accuracy, stability and conditioningof the computed solutions. The proposed framework is independentof any numerical method and therefore it can be used along withvarious possible numerical methods for the GQME solution, especiallymatrix flow-based algorithms (Chu, 1994) (e.g. continuation/homotopy,Morgan, 1989).