Stability Analysis for a Class of Partial Differential Equations Via

Home , Exponential stability, Lyapunov stability

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 inequality formulated by the proposed method becomes a polynomial matrix inequality. We then exploit the semidef- Abstract—This paper studies scalar integral inequalities 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 ΣP [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))) .  2 4  θ−1 −1 −1  g (x) 0 0 g (x) 0 0  h i  3 4  For max(deg(fb),deg(fi)) = k, we can express the polyno- However, a complete quadratic representation of the in- mials fb and fi as in the quadratic-like forms tegral expression (4), must also account for the differential relation among the entries of vθ(u) and is characterized by k T k b 2 b 2 b fb(vθ−1)= η⌈ ⌉(vθ−1(u)) Fbη⌈ ⌉(vθ−1(u)) (1) the set as in (7). Notice that the set is obtained by a straightforwardI application of the fundamental theorem T k k of calculus, which allows us to introduce matrices Hb fi(x, vθ(u)) = η⌈ 2 ⌉(vθ(u)) Fi(x)η⌈ 2 ⌉(vθ(u)) (2) related to the values at the boundary and matrix functions H x 1 σ(2nθ, k )×σ(2nθ, k ) i( ) (Ω). S 2 2 ∈C with symmetric matrix Fb ⌈ ⌉ ⌈ ⌉ and The example below illustrates matrices H and H (x) ∈ σ nθ, k σ nθ, k b i S ( 2 )× ( 2 ) symmetric matrix function Fi :Ω ⌈ ⌉ ⌈ ⌉ . for an element of set (7). The dependent variable u(x) is assumed→ to belong to sets Example 2. Consider (4) with fb = 2u(1)∂xu(0), of the form 2 2 fi(x, (u, ∂xu)) = sin (x)u∂xu + 2u∂xu, yielding k = 2, 2 (B) := u θ−1(Ω) Bvb =0 , (3) θ = 2, and vθ(u) = (u, ∂xu, ∂xu). Since the expres- B ∈C | θ−1 sion is homogeneous of degree k = 2, we replace the k Rnb×2nθ 2 with B , where nb is the number of constraints inhomogeneous vector η⌈ ⌉(vθ(u)) by a homogeneous vec- ∈ { k } on the boundary. tor (vθ(u)) 2 to obtain the quadratic expressions with b {1} {1} We study the following problem (vθ−1) = (u(1), ∂xu(1),u(0), ∂xu(0)), (vθ(u)) = 2 (u, ∂xu, ∂xu). The representation (2) is defined by Problem 1. Check whether the integral inequality 0 0 0 1 sin2(x) 0 − 2 1 b 0 0 0 0 2 x fb(vθ−1(u)) + fi(x, vθ(u)) dx 0, (4) Fb =   , Fi(x)=  − sin ( ) 0 0  ZΩ ≥ 0 0 0 0 2 1 0 0 0 1 0 0     holds for all u θ satisfying u (B).     ∈ H ∈B 3

⌈ k ⌉ b T ⌈ k ⌉ b ⌈ k ⌉ T ⌈ k ⌉ d ⌈ k ⌉ T ⌈ k ⌉ I := (η 2 (vθ− )) (Fb + Hb)η 2 (vθ− )+ (η 2 (vθ (u))) Fi(x)η 2 (vθ (u)) + (η 2 (vθ− (u)))) Hi(x)η 2 (vθ− (u)) dx 1 1 dx 1 1 ZΩ T 1 k b T k b k k ⌈ 2 ⌉ ⌈ 2 ⌉ ⌈ 2 ⌉ ⌈ 2 ⌉ | Fb ∈Fb(k,θ), Fi ∈Fi(k,θ), (η (vθ−1)) Hbη (vθ−1)+ η (vθ−1(u(x))) (Hi(x))η (vθ−1(u(x))) = 0 (7) x=0 ) d d 1 dx h11(x) dx h12(x)+ h11(x) 2 h12(x) d T T d d (vθ− (u)) Hi(x)(vθ− (u)) =(vθ ) h (x)+ h (x) h (x)+ h (x) h (x) (vθ) (8) x 1 1 dx 12 11 dx 22 12 22 d  1 h x h x  2 12( ) 22( ) 0   and the terms characterising the multiplicity of the integral Following the deﬁnition of set in (7) and the deﬁnition I as described by (7) are obtained with (8) and with of H¯i in (9) we obtain

k¯ b T k¯ b b T b b T −Hi(1) 0 b φ(u) = (η (v )) (Fb + Hb)η (v ) (vθ− ) Hb(vθ− )=(vθ− ) (vθ− ). θ−1 θ−1 1 1 1 0 Hi(0) 1 k¯ T k¯ + (η (vθ(u))) Fi(x)+ H¯i(x) η (vθ(u))dx. (12) Z Remark 1. Further to the non-uniqueness associated to Ω the algebraic relations in the vector describing the quadratic Hence, if the boundary term satisfies (10), and the integral term satisfies (11) then φ(u) 0 u (B). representation, which is characterised by the sets b, i, F F ≥ ∀ ∈B the Fundamental Theorem of Calculus exposes the non- Remark 2. Inequality (11) is a differential matrix in- uniqueness of the integral expression associated to the dif- equality since the elements H¯i(x) involve continuously ferential relations of the elements in vθ(u), characterising differentiable functions and their derivatives. the set (7). In order to simplify the presentation of the next result, A. Semidefinite programming formulation let us introduce the function H¯i(x), which satisfies Whenever matrix Fi(x) is a polynomial on variable x and we impose polynomial dependence of H¯i(x) on vari-

d k T k able x, inequality (11) can be addressed by a straightfor- (η⌈ 2 ⌉(vθ−1(u))) Hi(x)η⌈ 2 ⌉(vθ−1(u)) ward application of Putinar’s Positivstellensatz [14, The- dx k T k orem 2.14]. Note that the set Ω = [0, 1], can be described = (η 2 (v (u))) H¯ (x)η 2 (v (u)) (9) ⌈ ⌉ θ i ⌈ ⌉ θ as the semi-algebraic set x ω(x) := x(1 x) 0 . { | − ≥ } n ×n and allows us to denote the quadratic form in the integrand Corollary 1. For Fi(x) + H¯i(x) d d [x], if there ∈ R of (7) in terms of matrix F (x)+ H¯ (x). The quadratic- exists N(x) ΣnM ×nM [x] such that i i ∈ like characterization of the integrand in terms of the nM ×nM Fi(x)+ H¯i(x) N(x)ω(x) Σ [x] (13) algebraic and the differential relations leads to conditions − ∈ for integral inequalities in terms of matrix inequalities as then (11) holds. follows: Remark 3. If the coefficients of Fi(x) and H¯i(x) depend Theorem 1. If there exist Fb b, Fi(x) i, satisfy- affinely in unknown parameters and the degree of N(x) 1 ∈ F ∈ F ing (1)-(2), and Hi(x) (Ω), yielding Hb as in (7) and is fixed, checking whether (13) holds can be cast as a ∈C H¯i(x) as in (9) such that feasibility test of a convex set of constraints, an SDP, whose dimension depends on the degree of Fi(x) + H¯i(x) and ¯ ¯ k b T k b N(x) and on the dimension of matrix Fi(x)+ H¯i(x) which (η (vθ−1)) (Fb + Hb) η (vθ−1) 0 u (B), (10) ≥ ∀ ∈B depends on the degree k and the order θ as in (1)-(2). The formulation of inequalities (10) and (11) is possible Fi(x)+ H¯i(x) 0 x Ω, (11) ≥ ∀ ∈ thanks to the application of the Fundamental Theorem where k¯ = k , then inequality (4) holds in the subspace of Calculus to characterize the set of quadratic-like rep- 2 resentations of an integral inequality, as described by the defined by (B). B set (7). The terms introduced in the integrand by matrix Proof. For given polynomials fb and fi satisfying k = Hi do not affect the value of the integral and allow for a max(deg(fb),deg(fi)) we can express an integral expresion test for positivity based on the positivity of the matrices as in (4), say φ(u), using the quadratic forms as defined in the quadratic-like representation. This is reminiscent in (1)–(2) with Fb b(k,θ), Fi(x) i(k,θ), which gives of the quadratic representation that is used in sum-of- ∈F ∈F squares when checking positivity of a polynomial. Also, b φ(u) = fb(vθ−1)+ Ω fi(x, vθ(u)) dx, for polynomial expressions, the algebraic relations in the k¯ b T k¯ b = (η (vθ−1)) RFbη (vθ−1) quadratic representation of integrand polynomials are here k¯ T k¯ defined in terms of functions, (see (6)) instead of scalars. + Ω(η (vθ(u))) Fi(x)η (vθ(u))dx. R 4

With the solution to (10) it is possible to verify inequalities semigroups on a Hilbert space is given in terms of the in subspaces as in (3), incorporating boundary values of existence of a positive operator satisfying a Lyapunov the dependent variables. equation. Such a theorem is based on the results in [6]. The extension of this result for more general spaces has III. Stability Analysis for Partial Differential been provided in [22]. The above Theorem establishes con- Equations ditions for the stability and uniform asymptotic stability This section recalls some results for the stability anal- of nonlinear semigroups. Please refer to [16] for conditions ysis of PDEs and formulates integral inequalities for the on nonlinear operators to generate semigroups of nonlinear convergence of norms of the solutions of PDEs with poly- contractions. The remaining of this section proposes a nomial dependence on the dependent variables. Results sufficient condition for the existence of Lyapunov function extending Lyapunov’s second method to PDE systems for PDEs defined by nonlinear polynomial operators. In can be found in [17], [20], [31] and [12]. The formulation what follows, we present the class of PDE systems and detailed in the latter is recalled below: Lyapunov functionals studied in this paper. Consider the following PDE system Definition 1. A nonlinear semi-group on a compact α q normed space C is a family of maps S(t) C C, t t ∂tu = F (x, D u), u(t0, x)= u0(x) (Ω) (15) { | → ≥ 0} ∈M⊂H such that α where q N0. Let F (x, D u)= u, where is a nonlin- • for each t t , S(t) is continuous from C to C, ∈ N N ≥ 0 ear (polynomial) operator defined on , a closed subset of • for each u C, the mapping t S(t)u is continuous, q M ∈ → (Ω). Continuous solutions to the PDE (15) exist in • S(0) is the identity on C, Hand are unique, provided generates a nonlinear semi-M • S(t)(S(τ)u)= S(t+τ)u for all u C and all t, τ 0. N ∈ ≥ group of contractions. Definition 2. Let S(t), t t0 be a nonlinear semi- The following theorem is a Lyapunov result for the q group on C and for any{ u C≥, let }Y (u)= S(t)u, t t exponential convergence of the norm of the solutions 0 H be the orbit through u. We∈ say u is an equilibrium{ point≥ if} to (15). Y (u)= u . 1 { } Theorem 3. Suppose there exist a functional V , R ∈ C An orbit Y (u) is stable if for any ǫ > 0, there exists with V (0) = 0, and scalars c1, c2, c3 >0 such that ∈ δ(ǫ) > 0 such that for all t t0, S(t)u S(t)v C < ǫ 2 2 ≥ k − k c1 u q V (u) c2 u q (16) whenever u v C <δ(ǫ), v C, where C is the norm k k ≤ ≤ k k defined onkC.− Ank orbit is uniformly∈ asymptoticallyk·k stable if dV (u) 2 it is stable and also there is a neighbourhood D = v C c3 u q (17) { ∈ | dt ≤− k k u v C < r such that S(t)u S(t)v C 0 as t , q k − k } 1 k − k → → ∞ then, the norm of the trajectories of (15) satisfy uniformly for v D . Similarly, it is exponentially stable H ∈ c3 if there exist ν,γ > 0 such that 2 c2 2 − (t−t0) u(t, x) u (x) e c1 (18) k kq ≤ c k 0 kq −ν(t−t0) 1 S(t)u S(t)v C γ u v Ce , k − k ≤ k − k where u0 = u(t0, x). for all t t0 and all u, v C. dV (u) ≥ ∈ c Proof. From (16), (17), one has dt 3 . V (u) c1 Definition 3. Let S(t), t t0 be a nonlinear semi- dV (u) ≤− { ≥ } dt d(ln(V (u))) group on C. A Lyapunov function is a continuous real- Since V (u) = dt , the integral in time of valued function V on C such that the above expression over [t0,t] yields ln(V (u(t))) c3 − ln(V (u(t0))) (t t0) which gives V (u(t)) d V (S(t)u) V (u) c1 c3 ≤ − − ≤ V (u) = lim − 0, (14) − c (t−t0) dt t→0+ t ≤ V (u(t0))e 1 . Finally (18) is obtained by applying for all u C. the bounds of (16) on the above inequality. ∈ Theorem 2. (Lyapunov Theorem for Nonlinear Semi- Consider candidate Lyapunov functionals of the form groups, [12, Theorem 4.1.4]), Let S(t), t t be a 0 1 q T q nonlinear semi-group, and let 0 be{ an equilibrium≥ } point V (u)= (D u) P (x)(D u) dx, P (x) > 0 x Ω. 2 ZΩ ∀ ∈ in C. Suppose V is a Lyapunov function on C which (19) 1 satisfies V (0) = 0, and V (u) α u C for α > 0 and 2 1 1 That is, V (u) = 2 u (q,P (x)), the squared P (x)-weighted ≥ k k q k k u C. Then, 0 is stable. In addition, if ∂tV (u) α2 u C -norm. The following lemma states the equivalence of ∈ ≤− k k H q for α2 > 0, then 0 is uniformly asymptotically stable. the weighted norm and the -norm. H For the proof of the above theorem, refer to [12, p.84]. Lemma 1. If P (x) > 0 x Ω, then the norms u ∀ ∈ k k(q,P ) In [5, Theorem 5.1.3], necessary and sufficient conditions and u q are equivalent. for a linear operator defining a PDE to be the infinitesi- k k mal generator of exponentially stable strongly continuous The proof consists of showing that the inequalities √λm u q u (q,P ) √λM u q hold, with λm > 0, 1 k k ≤ k k ≤ k k i.e. ∀ε> 0, ∃T > 0: t > T s.t. kS(t)u − S(t)vkC < ε, ∀v ∈ D. λM > 0 satisfying λmI P (x) λM I, x Ω. ≤ ≤ ∀ ∈ 5

q1 Remark 4. For q1 < q2, the space is embedded in q H q 2 [8, Sec 5.6]. Therefore, stability in 2 -norm implies 1 H H stability in q1 -norm, but the converse does not necessarily 0.8 hold. H 0.6 Proposition 1. If there exists a function P (x) and positive p ( x) 0.4 scalars ǫ1, ǫ2 such that 0.2 q T q q T q (D u) P (x)(D u) ǫ1(D u) (D u) dx 0 (20) ZΩ − ≥ 0 0 0.2 0.4 0.6 0.8 1 x (Dqu)T P (x)F (x, Dαu)+ F T (x, Dq u)P (x)(Dqu) Fig. 1: Weighting functions proving exponential stability − ZΩ for convergence rates λ 2, 10 . The red dotted curves ∈ { } +ǫ (Dqu)T (Dqu) dx 0 (21) depict the solution p(x) = e−λx. The solid blue lines 2 ≥ correspond to the polynomials obtained by solving (24). then the q norm of solutions to (15) satisfy (18) with H c1 = minΩ(λmin(P (x))), c2 = maxΩ(λmax(P (x))), and c3 = ǫ2. 1 λp(x) d h(x) (h(x) p(x)) The verification of inequalities (20), (21) is a sufficient M(x)= − − dx − 0 2 (h(x) p(x)) 0 ≥ condition to verify the conditions of Theorem 3, since it − x Ω. (23c) restricts the class of functions in (16), (17) to be as in (19). ∀ ∈ Inequalities (20), (21) are integral inequalities such as Solve (23c) by imposing h(x) = p(x) and by solving the ones studied in Section II. The sets (B) as in (3) the differential equation ∂ h(x)+−λh(x) = 0, to obtain associated to the inequalities are defined byB the domain of x h(x) = e−λx, which satisfies (23a) (i.e. h(x)= p(x) > the PDE operators. The results of Section II-A can there- 0). Notice− that with p(x) = h(x) = e−λx− the inequality fore be applied to (20)-(21) whenever the integrands are of (23c) holds for all x R. Inequality− (23b) is expressed as polynomial functions. In this context, P (x) is a variable h(1)u2(1) > 0, which∈ clearly holds since h(1) = e−λ > parameterized in terms of polynomial coefficients. For such 0.− The inequalities then hold for any λ > −0 which proves a parameterization a tradeoff must be accounted for: as the exponential stability of the norm of the solution the class of parameterized functions is increased by adding 2 for any convergence rate. This isL expected as, for bounded elements elements of the basis (increasing the degree of domains, the equation presents finite-time stability. a polynomial P (x))the computational burden required to The inequalities in (20)-(21) were also formulated with solve (13) increases. polynomial weighting function p(x), with q = 0 (giving q D u = u) and ǫ2 = λ. Theorem 1 is applied to the resulting IV. Examples inequalities and we use the Positivstellensatz to formulate In this section we present solutions to inequali- the SOSP as in Corollary 1 ties (20), (21) by solving the SOS constraints obtained × M(x)+ N(x)x(x − 1) ∈ Σ2 2[x], Find p,h,N subject to × with the formulation of Corollary 1, (the constraints are N(x) ∈ Σ2 2[x]. cast as SDP with SOSTOOLS [18] and solved with Se- (24) DuMi [26]). The exponential stability of the norm of the L2 Solutions to the above inequalities were obtained for λ solution of a simple hyperbolic equation is studied below. (0, 10] (the value λ∗ = 10 was solved with deg(p(x)) =∈ Example A) Consider the equation deg(h(x)) = 30). The numerical results provide polynomial Lyapunov certificates for the 2 stability of the ∂tu = ∂xu x [0, 1], t> 0 u(t, 0)=0, (22) L − ∈ solutions of (22). A comparison of the solution p(x)= e−λθ u(1) 2 0 1 and the numerical solutions are depicted in Figure 1. which gives = u (Ω) [ ] u(0) =0 . Given B n ∈C | h i o In the next example we study the stability of a nonlin- 1 2 λ > 0, let Ep = p(x)u (x) dx, be the candidate ear, inhomogeneous PDE. 2 Ω R 1 2 Example B) (System of nonlinear inhomogeneous PDEs) function to certify λEp E˙ p = λp(x)u (x) + − − Ω −2 Consider the following PDE p(x)u∂xu(x) dx 0 (proving the exponentialR stability ≥ −1 2 with convergence rate λ). We apply Theorem 1 to the ∂tu = R u + C(x)u + u (D(x) u) (25) Lyapunov inequalities in Proposition 1 to obtain ∇ × ∇ where u( , x): R R3, R> 0 and p(x) > 0 x Ω (23a) · → ∀ ∈ 0 0 0 00 0 T u(1) h(1) 0 u(1) u(1) C(x)=  x 0 0  ; D(x)=  00 0  − > 0 u(0) 0 h(0) u(0) ∀ u(0) ∈B x x2 0 0 0 1 + x  −   − 2  (23b) (26) 6

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