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Stability Analysis for a Class of Partial Differential Equations Via 1 Stability Analysis for a Class of Partial terms of differential matrix inequalities. The proposed Differential Equations via Semidefinite method is then applied to study the stability of systems q Programming of inhomogeneous PDEs, using weighted norms (on Hilbert spaces) as LF candidates. H Giorgio Valmorbida, Mohamadreza Ahmadi, For the case of integrands that are polynomial in the Antonis Papachristodoulou independent variables as well, the differential matrix in- equality formulated by the proposed method becomes a polynomial matrix inequality. We then exploit the semidef- Abstract—This paper studies scalar integral inequal- ities in one-dimensional bounded domains with poly- inite programming (SDP) formulation [21] of optimiza- nomial integrands. We propose conditions to verify tion problems with linear objective functions and poly- the integral inequalities in terms of differential matrix nomial (sum-of-squares (SOS)) constraints [3] to obtain inequalities. These conditions allow for the verification a numerical solution to the integral inequalities. Several of the inequalities in subspaces defined by boundary analysis and feedback design problems have been studied values of the dependent variables. The results are applied to solve integral inequalities arising from the using polynomial optimization: the stability of time-delay Lyapunov stability analysis of partial differential equa- systems [23], synthesis control laws [24], [29], applied to tions. Examples illustrate the results. optimal control law design [15] and system analysis [11]. Index Terms—Distributed Parameter Systems, In the context of PDEs, preliminary results have been PDEs, Stability Analysis, Sum of Squares. presented in [19], where a stability test was formulated as a SOS programme (SOSP) and in [27] where quadratic integrands were studied. I. Introduction The paper is organised as follows. Section II defines The need for accurate models to study dynamical sys- the problem, illustrates how we can write polynomial tems [1], [2], [4], [30] have driven research efforts towards integrands in quadratic-like forms and proposes a method partial differential equation (PDE) systems - equations to verify integral inequalities by studying matrix inequal- involving derivatives with respect to more than one inde- ities associated with quadratic-like representation of the pendent variable. Such infinite dimensional system models integrands. In Section III, some results for the stability are challenging to study both analytically and numerically. analysis of PDE systems are summarised. Two examples Conventional numerical approaches to study PDEs rely on illustrate the application of the proposed method to the spectral or spatial discretization and use tools developed stability analysis of PDEs in Section IV. Finally, Section V for Ordinary Differential Equations (ODEs) [10], [7]. Nu- concludes the paper and gives directions for future re- merical methods which do not require finite-dimensional search. approximations are needed may prevent conservatism in R R R Rn the system analysis. Notation. Let , ≥0, >0 and denote the field Stability analysis of PDE systems can be performed with of reals, non-negative reals, positive reals and the n- the Lyapunov’s second method, which was extended to dimensional Euclidean space, respectively. The sets of infinite dimensional systems in [6], [17]. One advantage natural numbers and positive natural numbers are denoted N N of this method is that it does not require calculation of 0, . The closure of set Ω is denoted Ω. The boundary ∂Ω of set Ω is defined as Ω Ω with denoting set semi-groups [16], but instead requires the construction \ \ of a Lyapunov Functional (LF). A priori choices for LF subtraction. The ring of polynomials, the ring of positive structures for a given system are difficult to make. In polynomials, and the ring of sum-of-squares polynomials on a real variable x are denoted [x], [x] and Σ[x] particular cases, for instance when the PDE contains R P energy-preserving non-linearities such as convection terms, respectively. The ring of SOS square matrices of dimension n×n taking the energy of the state as the LF simplifies the n, i.e., matrices M(x) [x] satisfying M(x) = nM ∈ R N T x N x N x Rdi×n stability analysis [25]. The Lyapunov conditions for PDE i=1 i ( ) i( ) with i( ) , is denoted n×n N ∈ Nn×σ(n,k) systems take the form of integral inequalities, therefore PΣ [x]. For n, k , define the matrix K , n+k−1∈ (n+k−1)! ∈ methods for computing the LFs require the solution of σ(n, k) := n−1 = (n−1)!k! , of which the columns n satisfy Kij =k, j, without repetition. The multi- integral inequalities. i=1 ∀ This paper presents a method to verify integral inequali- index notationP is used to define the vector of all monomials N Rn ties with integrands given by polynomials in the dependent of degree k on vector w = (w1, w2,...,wn) , as {k} ∈K · T K · T ∈ K · variables. The polynomial structure allows for a quadratic- w := (w ( 1) ) (w ( σ(n,k) ) where w ( j) = n Kij ··· {k} like representation of the integrand and we formulate i=1 wi . The number of terms in w is hence given conditions for the positivity of integral expressions in Qby σ(n, d). For instance, with n = 2 and k = 2 one 2 1 0 {2} 2 2 has K = [ 0 1 2 ], and w = (w1, w1w2, w2). Define The authors are with the Department of Engineering Science, the vector containing all monomials in w up to degree University of Oxford, Parks Road, Oxford, OX1 3PJ, UK e-mail: ({ T giorgio.valmorbida, mohamadreza.ahmadi, antonis }@eng.ox.ac.uk), k as ηk(w) := 1 (w{1})T (w{k})T . The set ··· tel +44 1865 2 83035. G. Valmorbida is also affiliated to Somerville of continuous vector functions, mapping Ω R into College, University of Oxford, Oxford, U.K. Work supported by Rn ⊂ EPSRC grant EP/J010537/1. M. Ahmadi is supported by the Oxford , which are k-times differentiable and have continuous Clarendon Scholarship and the Sloane-Robinson Scholarship. derivatives is denoted k(Ω). For p(x) 1(Ω), the d- C ∈ C 2 th derivative of p with respect to variable x is denoted For a given polynomial fb, the representation (1) may d α N α ∂xp. For u (Ω), α 0, define D u = vα(u) := be non-unique and is taken as an element of the set α∈ C ∈ α (u, ∂xu,...,∂x u), α is the order of D u. The variable u is σ nθ, k σ nθ, k S ( 2 )× ( 2 ) called the dependent variable and x Ω, the independent b(k,θ)= Fb + Gb : ⌈ ⌉ ⌈ ⌉ variable. For notational convenience,∈ depending on the F n k b T k b fi = (η 2 (v )) Fbη 2 (v ), context we may suppress the dependence of u(t, x) on | ⌈ ⌉ θ−1 ⌈ ⌉ θ−1 k k t (by writing u(x)) or x (by writing u(t)) or both (by 2 b T 2 b 0 = (η⌈ ⌉(vθ−1)) Gbη⌈ ⌉(vθ−1) . (5) writing u). The set of functions satisfying u α(Ω) for o ∈ C which u 2 = (Dαu(x))T (Dαu(x)) dx < is denoted Similarly, for a given function f , the set of quadratic-like k kα Ω ∞ i α(Ω). For aR matrix P (x) > 0 x Ω, define the representation (2) is taken as an element of the set 1 H α T∀ ∈ α 2 norm u (α,P (x)) := (D u(x)) P (x)(D u(x)) dx . Ω σ nθ, k σ nθ, k k k n S ( 2 )× ( 2 ) The set of real symmetricR matrices is denoted S = A i(k,θ)= Fi(x)+ Gi(x):Ω ⌈ ⌉ ⌈ ⌉ Rn×n T Sn { ∈ F n → A = A . For A , denote A 0 (A > 0) if k T k | } ∈ ≥ fi = (η 2 (vθ(u))) Fi(x)η 2 (vθ(u)), A is positive semidefinite (definite). The ceil function is | ⌈ ⌉ ⌈ ⌉ k T k denoted . The Jacobian of a vector field is denoted by 2 2 0 = (η⌈ ⌉(vθ(u))) Gi(x)η⌈ ⌉(vθ(u)) . (6) , the Laplacian⌈·⌉ is denoted 2 and the cross-product is o denoted∇ by the symbol . ∇ The following example illustrates the above definitions. × 2 2 Example 1. Consider fi(x, (u, ∂xu)) = x ∂xu+2u(∂xu) , II. Integral inequalities with polynomial u u u(0) = u(1) , yielding k =3, θ = 1, and vθ(u)= ∈{ | k } 2 2 integrands (u, ∂xu), η⌈ 2 ⌉(vθ(u)) = (1, u, ∂xu,u , u∂xu, (∂xu) ). The quadratic representation (2) and the set (3) are obtained In this section, we study inequalities given by polynomi- b u(1) u(0) with vθ−1 = [ ], als on the dependent variables evaluated at the boundaries x2 of the domain of integration and integral terms with 0 0 2 0 0 0 polynomial integrands on the dependent variables. 0 0 0 001 x2 Consider the integral inequality Fi(x)= 2 0 0 000 , B = 1 −1 . 0 0 0 000 0 0 0 000 fb(vθ−1(u(1)), vθ−1(u(0))) + fi(x, vθ(u(x))) dx 0, 0 1 0 000 ZΩ ≥ with Ω = [0, 1], fb [vθ (u(1)), vθ (u(0))], fi( , vθ) The set i(k,θ) is defined by matrices Gi(x) as ∈ R −1 −1 · ∈ F [vθ] (fi is a polynomial on its second argument). In order R 0 0 0 g1(x) g2(x) g3(x) to simplify the exposition, let us define 0 −2g1(x) −g2(x)0 0 0 0 −g2(x) −2g3(x)0 0 0 Gi(x)= . T g1(x)0 0 0 0 g4(x) b T T g (x)0 0 0 −2g (x) 0 v (u) := (vθ (u(1))) (vθ (u(0))) .
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