Hierarchical Stability Notions and Lyapunov Functions for PDEs

Hierar chical Stability Notions and L yapunov Functions f or PDEs Matthe w M. Peet Abstract — Unlike linear ordinary differential equations (ODEs), linear partial differential equations (PDEs) admit a multitude of non-equiv alent notions of stability . This variety makes interpr etation of L yapunov stability results challenging. T o simplify this inter pretation, we propose a framework for hierarchical classification of notions of stability and L yapunov conditions. T o do this, for every well-posed PDE and set of boundary conditions, we define a fundamental state on L 2 corresponding to the minimal information needed to uniquely forward propagate the solution. Stability notions and L yapunov functions are then defined in terms of this fundamental state. This gives rise to a hierarch y of stability notions, the weakest being fundamental state to PDE state stability . Other stability notions and L yapunov conditions may then be interpreted relati ve to this weakest notion. Hierarchies are established for: L yapunov , exponential and finite-energy stability . Sufficient L yapunov conditions are defined in terms of operator inequali- ties. Illustrative examples and computational tools ar e provided. I . I N T RO D U C T I O N Recent years hav e seen an increase in methods for sta- bility analysis and control of partial differential equations (PDEs) without discretization. Often, these methods rely on L yapunov functions to establish con ver gence of the system state, bound L 2 gain, or design observers and controllers. Unlike ODEs, for which we hav e a well-established univ ersal state-space representation, PDEs take a variety of forms – with no clear and consistent notion of state space. As a result, basic definitions of stability vary from PDE to PDE and, ev en for a giv en PDE, will vary with the set of im- posed boundary conditions 1 . The use of L yapunov functions further complicates the problem. While it is accepted that a L yapunov function should be positiv e, bounded, and non- increasing, the form taken by these upper and lo wer bounds is not consistent. As a result, even for a giv en PDE with a given set of boundary conditions, we may obtain a v ariety of non- equiv alent stability results depending on ho w the L yapunov function is defined and bounded. T o illustrate, consider a well-studied PDE: the wav e equa- tion u tt = u xx with pinned boundary conditions u ( t, 0) = u ( t, 1) = 0 . This PDE is not stable in the L 2 -norm sense in that there exists no C ≥ 0 such that ∥ u ( t ) ∥ L 2 + ∥ ˙ u ( t ) ∥ L 2 ≤ C ( ∥ u (0) ∥ L 2 + ∥ ˙ u (0) ∥ L 2 ) for all t ≥ 0 . Howe ver , this PDE is stable in another sense, in that for V ( u ) = ∥ ˙ u ∥ L 2 + ∥ u s ∥ L 2 , ˙ V = 0 and hence ∥ u s ( t ) ∥ L 2 + ∥ ˙ u ( t ) ∥ L 2 ≤ ∥ u s (0) ∥ L 2 + ∥ ˙ u (0) ∥ L 2 . Howe ver , this tells us nothing about the actual M. Peet is with the School for the Engineering of Matter , Transport and Energy , Arizona State University . e-mail: mpeet@asu.edu This work was supported by the National Science Foundation under grants No. 2429973 and 2337751. 1 Semigroup theory provides basic definitions of state which apply to a large class of PDEs. Howe ver this approach simply shifts the problem into various definitions of inner product, domain, generator , et c. PDE state, u ( t ) . Fortunately , we know that ∥ u ∥ ≤ C ∥ u s ∥ and hence ∥ u ( t ) ∥ L 2 + ∥ ˙ u ( t ) ∥ L 2 ≤ C ∥ u s (0) ∥ L 2 + ∥ ˙ u (0) ∥ L 2 for some C > 0 . Howe ver , as indicated, this bound cannot be strengthened to the first, more tradition notion of stability or well-posedness in u . This illustration shows that obvious no- tions of stability often do not hold for PDEs (beam equations hav e the same problem). Nonetheless, it seems reasonable that we would want to be able to include wav e equations in any L yapunov framework for analysis and control of PDEs. Clearly , then, we must allow for some weaker notions of stability in the analysis and control of PDEs. The need for a weaker notion of stability (and well- posedness) is well-kno wn. Such notions for the wav e equa- tion can be found in [1] and other e xamples where stability depends on the choice of norm can be found in [2]. For a broader class of PDE, the framework dev eloped in [3]–[5] for non-coerciv e L yapunov functions weakens the positivity condition V ( u ) ≥ ϵ ∥ u ∥ 2 to V ( u ) > 0 for all u  = 0 . A strict negativity condition on the deriv ativ e of the L yapunov function is then used to upper bound the solutions. This is similar the use of storage functions. Howe ver , the difficulty in proving stability of PDE’ s such as the wav e equation lies not in positi vity of the L yapunov function, but in the upper bound. For e xample the L yapunov function used for the w av e equation discussed abo ve is coercive in ∥ u ∥ . Howe ver , it is not upper bounded by ∥ u ∥ . The contribution of this paper is relativ ely modest. W e simply classify se veral properties of L yapunov functions typically used in analysis of PDEs. These properties are divided into 3 categories: positivity (lower bounded), upper bounded, and ne gativity of the deri vati ve. W e then interpret those properties in terms of a hierarchy of notions of stability . The dif ficulty , as suggested abo ve, is that the PDEs, boundary conditions, and L yapunov functions v ary substantially from case to case. T o address this problem, the frame work pro- posed in this paper is structured around the Partial Integral Equation (PIE) or fundamental state representation of PDEs. Specifically , if the boundary conditions of a PDE are suitably well-defined, there exists a bijection from L 2 to the set of sufficiently regular functions which satisfy the associated set of boundary conditions. For example, consider the heat equation u t = u ss where solutions are constrained to lie in some domain, say X := { u ∈ H 2 : u (0) = u (1) = 0 } . Then if we define T as ( T x ) := Z s 0 θ ( s − 1) x ( θ ) dθ + Z 1 s s ( θ − 1) x ( θ ) dθ W e hav e that T : L 2 → X , T ∂ 2 s u = u and ∂ 2 s T x = x for any u ∈ X and x ∈ L 2 . W e refer to x := u ss as the fundamental state. Using this bijection, the dynamics of the PDE may be equiv alently 2 written as T ˙ x = A x . Of course, for the heat equation, this is simply 3 T ˙ x = x and T = ( ∂ 2 s ) − 1 on the domain X . An equation of the form T ˙ x = A x where T and A have the structure given above is a Partial Inte gral Equations (PIEs). An operator T of this type is a Partial Integral (PI) operator and the set of PI operators define a linear algebra, with the set of PI operators with polynomial kernels being a sub-algebra. It has been shown that any PDE with suitably well-defined domain, X , admits such a representation with polynomial kernels [6]–[8]. Our approach, then, is to allow for classification of bounds on a L yapunov function in terms of both the PDE state and the fundamental (PIE) state. Next, we classify notions of stability in terms of these bounds – focusing on two cases of L yapunov stability: PDE stability ( ∥ u ( t ) ∥ ≤ C ∥ u (0) ∥ ) and the weaker notion of PIE to PDE stability ( ∥ u ( t ) ∥ ≤ C ∥ x (0) ∥ ). These notions are extended to exponential and finite-energy stability . This classification system then allo ws for interpretation of almost any PDE stability result which is based on the use of L yapunov functions. Finally , to illus- trate the application of these results, we express sufficient conditions for these v arious stability notions in terms of operator inequalities. These operator inequalities are applied to representativ e PDEs. I I . F U N DA M E N T A L ( P I E ) S TA T E A N D B I J E C T I O N T Definition 1: W e say P is a PI operator , denoted P ∈ Π 3 ⊂ L ( L 2 ) if ( P x ) := R 0 ( s ) x ( s )+ Z s 0 R 1 ( s, θ ) x ( θ ) dθ + Z 1 s R 1 ( s, θ ) x ( θ ) dθ where R 0 ∈ L ∞ and R 1 , R 2 ∈ L 2 . Furthermore, we define Π 2 := { P ∈ Π 3 : R 0 = 0 } . Π 3 and Π 2 are vector spaces and, when the R i are square, linear *-algebras under composition and L 2 adjoint. The subspaces of Π 3 and Π 2 with R i polynomial are like wise subalgebras. Furthermore, for P ∈ Π 3 and Q ∈ Π 2 , we hav e PQ , QP ∈ Π 2 . Extension to multiv ariate domains is the sum and composition of Π 2 , Π 3 operators on each spatial dimension. Extension to mixed spatial domains such as L ( R n × L 2 [0 , 1] × L 2 [[0 , 1] 2 ]) can be found in [7], [8]. PDEs are typically defined in terms of ˙ u = A u , where A is a differential operator and u is constrained to lie in some subspace of a Sobolev space – which we refer to as X . If the PDE is well-posed, then A : X → L 2 is inv ertible on X and T := A − 1 ∈ Π 3 . This results in an equation of the form T ˙ x ( t ) = x ( t ) where we refer to x ( t ) ∈ L 2 as the fundamental state. Howe ver , solving for A − 1 explicitly requires solution of certain dif ferential equations. In [6], a framew ork was proposed whereby , roughly speaking, only the highest deriv ativ e, D α , in A is inv erted 4 . This results in 2 For notational simplicity , we assume sufficient regularity for commuta- tion of spatial and temporal deriv atives. Otherwise we use ∂ t ( T x ) = A x . 3 More generally , we may always write the dynamics of a PDE ˙ u = Au as T ˙ x = x where T = A − 1 on X . Howev er, in this case, the kernels of operator T may not be polynomial and require analytic solution of certain differential equations. If such an operator is known, howev er, the results of this paper can be applied using this T and A = I . 4 If A is not a dif ferential operator , boundary conditions still require differentiation in order to obtain a bijection from L 2 to X . a Partial Integral Equation (PIE) of the form T ˙ x ( t ) = A x ( t ) where T = ( D α ) − 1 : L 2 → X and A = A T ∈ Π 3 . For this case, analytic formulae can be obtained for the polynomial kernels in T . This is illustrated in the following lemma which concisely states the result in [6] and where we use D n = ∂ n s , X = n g ∈ H n [ a, b ] : for i = 1 , · · · , n (1) n X j =1 α i,j ( D j − 1 g )( a ) + n X j =1 β i,j ( D j − 1 g )( b ) = 0 o and W n ( t ) =         1 t 1 2 t 2 · · · t n − 1 ( n − 1)! 0 1 t · · · t n − 2 ( n − 2)! . . . . . . . . . . . . . . . 0 0 0 · · · t 0 0 0 · · · 1         . Lemma 1: F or X, α i,j , β i,j as defined in Eq. (1), [ N a ] i,j = α i,j , [ N b ] i,j = β i,j if det ( N a + N b W ( b − a ))  = 0 , let H := D n and ( T x )( s ) = Z b a G ( s, θ ) x ( θ ) dθ G ( s, θ ) = ( e 1 ( s − a ) T ( I − P ) e n ( a − θ ) θ < s − e 1 ( s − a ) T P e n ( a − θ ) s < θ where P = ( N a + N b W ( b − a )) − 1 N b W ( b − a ) and e 1 ( s ) =        1 s 1 2 s 2 . . . s n − 1 ( n − 1)!        , e n ( θ ) =        θ n − 1 ( n − 1)! . . . 1 2 θ 2 θ 1        . Then for any u ∈ X and x ∈ L 2 , we have HT x = x , TH u = u and T x ∈ X . W e refer to u as the PDE state and x = D α u as the fundamental or PIE state. Stability notions and L yapunov functions may then be defined in terms of the PIE state and, occasionally , interpreted in terms of the PDE state, u = T x . I I I . P O S I T I V I T Y , B O U N D E D N E S S , A N D N E G A T I V I T Y For each notion of stability which will be introduced in Sec. IV, we will associate a L yapunov stability condition. Each of these L yapunov conditions includes three parts: a lower bound on the function (positivity), an upper bound on the function, and a bound on the negati vity of the deriv ative of the function. Before introducing notions of stability , we will characterize useful v ariations on these 3 properties. Specifically , we list possible properties of a L yapunov func- tion, V , in terms of PIE state, x = D α u where the PDE state is obtained from the PIE state as u = T x for some PI operator , T . This then implies that ∥ u ∥ L 2 ≤ ∥ T ∥∥ x ∥ . W e start with positivity properties, from weakest to strongest. Definition 2 (Lyapuno v positivity): For a given operator T : L 2 → X and L yapunov function, V : L 2 → R + with V (0) = 0 , we say that V is 1) Positi ve semidefinite if V ( x ) ≥ 0 . 2) PDE positiv e if V ( x ) ≥ ϵ ∥ T x ∥ 2 for some ϵ > 0 . 3) PIE positiv e if V ( x ) ≥ ϵ ∥ x ∥ 2 for some ϵ > 0 . These conditions are required to hold for all x ∈ L 2 . Note 1: Note that we might have added that V is Positi ve definite if V ( x ) > 0 for all x  = 0 – i.e. positivity in the non- coerciv e sense. Howe ver , this definition is somewhat dif ficult to interpret constructiv ely . One possible mechanism is to say that V is positi ve definite if V ( x ) ≥ ϵ ∥ R x ∥ 2 for some in vertible mapping R – so that PDE positivity then becomes a special case with R = T . This is similar in spirit to the non-coerciv e function proposed for the heat equation in [4] which can be expressed as V ( x ) = ⟨ TT x , TT x ⟩ = ∥ TT x ∥ 2 . The positivity types in Defn. 2 are hierarchical – PIE pos- itiv e implies PDE positiv e implies positi ve definite implies positiv e semidefinite. The first holds because V ( x ) ≥ ϵ ∥ x ∥ 2 and ∥ T x ∥ ≤ ∥ T ∥∥ x ∥ imply V ( x ) ≥ ϵ ∥ x ∥ 2 ≥ ϵ ∥ T 2 ∥ ∥ T x ∥ 2 . Howe ver , PDE positivity does not imply PIE positivity , since there is no C such that ∥ x ∥ ≤ C ∥ T x ∥ . For upper bounds, from weak to strong, we ha ve Definition 3 (Lyapuno v bounds): Gi ven a L yapunov func- tion, V : L 2 → R + with V (0) = 0 , we say that V is 1) PIE bounded if V ( x ) ≤ C ∥ x ∥ 2 for some C > 0 . 2) PDE bounded if V ( x ) ≤ C ∥ T x ∥ 2 for some C > 0 . Here the inequalities must hold for and all x ∈ L 2 . Any L yapunov function of the form V ( x ) = ⟨ x , P x ⟩ with PI operator P ∈ Π 3 is PIE bounded. Furthermore, PDE bounded implies PIE bounded since ∥ T x ∥ ≤ ∥ T ∥∥ x ∥ . For negativity , we have the follo wing Definition 4 (Lyapuno v Derivative Conditions): Gi ven a L yapunov function, V : L 2 → R + with deriv ativ e ˙ V ( x ) where ˙ V (0) = 0 , we say that ˙ V is 1) Negative semidefinite if ˙ V ( x ) ≤ 0 2) PDE negativ e if ˙ V ( x ) ≤ − α ∥ T x ∥ 2 3) PIE negativ e if ˙ V ( x ) ≤ − α ∥ x ∥ 2 4) L yapunov Negative if ˙ V ( x ) ≤ − αV ( x ) for some α > 0 . Inequalities must hold for and all x ∈ L 2 . Apart from “L yapunov negati ve”, negativity conditions are ordered from weakest to strongest – i.e. PIE negati ve implies PDE neg ativ e implies negati ve semidefinite. The first holds since −∥ x ∥ 2 ≤ − 1 ∥ T ∥ 2 ∥ T x ∥ 2 . The strength of “L yapunov negati vity” relativ e to PIE and PDE negati vity depends on positivity and boundedness of V . Specifically , L yapunov negati vity implies PDE or PIE negativity if V is PDE or PIE positi ve, respectiv ely . Like wise, PDE or PIE negati vity implies L yapunov negati vity if V is PDE or PIE, respecti vely . I V . S T A B I L I T Y N O T I O N S A N D L Y A P U N OV F U N C T I O N S In this section, we define 4 notions of stability and, for each notion of stability , associate a sufficient condition in terms of existence of a L yapunov function which satisfies a set of bounds as defined in Sec. III. For consistency , these notions of stability and L yapunov conditions are all expressed in terms of the fundamental state, x ∈ L 2 . This formulation simplifies the presentation and the toolset used to verify the proposed L yapunov conditions – since x ∈ L 2 is not differentiable. When a condition in volv es the term T x ∈ X , this is interpreted as the state of the underlying PDE from whence the fundamental state was obtained and hence is restricted by boundary conditions and continuity constraints. A. Notions of L yapunov Stability Let us begin with the notions of L yapunov stability . These notions are particularly important for energy-conserving sys- tems such as wave and beam equations. Definition 5: W e say T ˙ x = A x is 1) L yapunov PIE to PDE stable if ∥ T x ( t ) ∥ ≤ C ∥ x (0) ∥ 2) L yapunov PIE stable if ∥ x ( t ) ∥ ≤ C ∥ x (0) ∥ 3) L yapunov PDE stable if ∥ T x ( t ) ∥ ≤ C ∥ T x (0) ∥ 4) L yapunov PDE to PIE stable if ∥ x ( t ) ∥ ≤ C ∥ T x (0) ∥ for some C > 0 and an y x ( t ) with T ˙ x ( t ) = A x ( t ) . While the notion of PDE to PIE stable is included for completeness, such a notion ne ver holds when the PIE state is a partial deriv ativ e of the PDE state ( x = D α u ) since at time t = 0 , this requires ∥ x (0) ∥ ≤ C ∥ T x (0) ∥ .In the following, we examine the relationships between the sev eral proposed notions of L yapunov stability . Lemma 2: Gi ven T ˙ x = A x , the following are true. 1) PIE stable implies PIE to PDE stable 2) PDE stable implies PIE to PDE stable 3) PDE to PIE stable implies PIE stable, PDE stable and PIE to PDE stable Pr oof: For statement 1), we ha ve ∥ x ( t ) ∥ ≤ C ∥ x (0) ∥ implies PIE to PDE stable since ∥ T x ( t ) ∥ ≤ ∥ T ∥∥ x ( t ) ∥ ≤ ∥ T ∥ C ∥ x (0) ∥ . For statement 2), we hav e ∥ T x ( t ) ∥ ≤ C ∥ T x (0) ∥ implies PIE to PDE stable since ∥ T x ( t ) ∥ ≤ C ∥ T x (0) ∥ ≤ ∥ T ∥ C ∥ x (0) ∥ . For statement 3), we have that ∥ x ( t ) ∥ ≤ C ∥ T x (0) ∥ implies PDE stable since ∥ T ( t ) ∥ ≤ ∥ T ∥∥ x ( t ) ∥ ≤ ∥ T ∥ C ∥ T x (0) ∥ . Similarly , PIE stable follows since ∥ x ( t ) ∥ ≤ C ∥ T x (0) ∥ ≤ ∥ T ∥ C ∥ x (0) ∥ . Statements 1) or 2) then imply PIE to PDE stable. Lemma 2 shows that PIE to PDE stable is the weakest notion, since it is implied by any of the others. Howe ver , PDE stable does not imply PIE stable and PIE stable does not imply PDE stable. a) Lyapunov Conditions for Lyapunov Stability: Hav- ing defined sev eral notions of L yapunov stability , we now provide conditions for such stability properties in terms of properties of a candidate L yapunov function. Lemma 3 (Conditions for L yapunov stability): Suppose there exists V : L 2 → R + with V (0) = 0 and with ˙ V ( x ) = lim h → 0 + V ( ˆ x ( t + h )) − V ( x ) h for any x ( t ) such that T ˙ ˆ x ( t ) = A ˆ x ( t ) with ˆ x (0) = x . Then T ˙ x = A x is 1) PIE to PDE stable if V is PDE positiv e, PIE bounded, and negati ve semidefinite. 2) PIE stable if V is PIE positi ve, PIE bounded, and negati ve semidefinite. 3) PDE stable if V is PDE positi ve, PDE bounded, and negati ve semidefinite. 4) PDE to PIE stable if V is PIE positiv e, PDE bounded, and negati ve semidefinite. Pr oof: In all cases, ˙ V ≤ 0 implies V ( x ( t )) ≤ V ( x (0)) . Hence, for statement 1, we have ϵ ∥ T x ( t ) ∥ 2 ≤ V ( x ( t )) ≤ V ( x (0)) ≤ C ∥ x (0) ∥ 2 . For statement 2, ϵ ∥ x ( t ) ∥ 2 ≤ V ( x ( t )) ≤ V ( x (0)) ≤ C ∥ x (0) ∥ 2 . For Statement 3, ϵ ∥ T x ( t ) ∥ 2 ≤ V ( x ( t )) ≤ V ( x (0)) ≤ C ∥ T x (0) ∥ 2 . For statement 4, ϵ ∥ x ( t ) ∥ 2 ≤ V ( x ( t )) ≤ V ( x (0)) ≤ C ∥ T x (0) ∥ 2 . B. Notions of Exponential Stability For exponential stability , from weak to strong, we hav e Definition 6: Gi ven a well-posed PIE of the form T ˙ x = A x , we say the PIE is 1) Exp. PIE to PDE stable if ∥ T x ( t ) ∥ ≤ C e − αt ∥ x (0) ∥ 2) Exp. PIE stable if ∥ x ( t ) ∥ ≤ C e − αt ∥ x (0) ∥ 3) Exp. PDE stable if ∥ T x ( t ) ∥ ≤ C e − αt ∥ T x (0) ∥ 4) Exp. PDE to PIE stable if ∥ x ( t ) ∥ ≤ C e − αt ∥ T x (0) ∥ for some C , α > 0 and an y x ( t ) with T ˙ x ( t ) = A x ( t ) . As in Lemma 2, Exp. PIE or PDE stable implies Exp. PIE to PDE stable and Exp. PDE to PIE stable implies all others. L yapunov Conditions for Exponential Stability T o keep the conditions tight and concise, in Lemma 4 we use L ya- punov negati vity instead of PIE or PDE negati vity . This allows us to directly use V ( x ( t )) ≤ − αV ( x ( t )) to conclude V ( x ( t )) ≤ e − αt V ( x (0)) . W e will return to PIE and PDE negati vity in Subsec. IV -C and Sec. V, while for now recall- ing only that the relation to L yapunov negativity depends on PIE and PDE positivity and boundedness. Lemma 4 (Conditions for Exponential stability): Suppose there exists V : L 2 → R + with V (0) = 0 and with ˙ V ( x ) = lim h → 0 + V ( ˆ x ( t + h )) − V ( x ) h for x ( t ) such that T ˙ ˆ x ( t ) = A ˆ x ( t ) with ˆ x (0) = x . Suppose ˙ V is L yapunov negati ve. Then T ˙ x = A x is 1) Exp. PIE to PDE stable if V is PDE positive, PIE bounded. 2) Exp. PIE stable if V is PIE positiv e, PIE bounded. 3) Exp. PDE stable if V is PDE positive, PDE bounded. 4) Exp. PDE to PIE stable if V is PIE positive, PDE bounded. Pr oof: F or statement 1, we have ϵ ∥ T x ( t ) ∥ 2 ≤ V ( x ( t )) ≤ e − αt V ( x (0)) ≤ C e − αt ∥ x (0) ∥ 2 . For statement 2, ϵ ∥ x ( t ) ∥ 2 ≤ V ( x ( t )) ≤ e − αt V ( x (0)) ≤ C e − αt ∥ x (0) ∥ 2 . For statement 3, ϵ ∥ T x ( t ) ∥ 2 ≤ V ( x ( t )) ≤ e − αt V ( x (0)) ≤ C e − αt ∥ T x (0) ∥ 2 . For statement 4, ϵ ∥ x ( t ) ∥ 2 ≤ V ( x ( t )) ≤ e − αt V ( x (0)) ≤ C e − αt ∥ T x (0) ∥ 2 . C. More Exotic Flavours: F inite Energy Stability One might interpret “non-coerciv e” L yapunov functions as V positive definite, but not PDE positiv e. Howe ver , we don’t hav e e xactly a consistent mechanism to enforce V positive definite and so we will relax this to V positive semidefinite. In order to ensure a non-zero L yapunov function, we then enforce a strict negati vity condition on the deriv ativ e. A con- sequence of this formulation is that the L yapunov function does not enforce a pointwise-in-time decay rate in the same way that exponential stability does, but rather only imposes a bound on the energy of the solution as measured by its temporal L 2 norm – i.e. ∥ x ∥ L 2 = q R ∞ 0 ∥ x ( t ) ∥ 2 dt . W e refer to this notion of ener gy as ”Finite-Energy” Stability . Definition 7: Gi ven a well-posed PIE of the form T ˙ x = A x , we say the PIE is 1) PIE to PDE FE stable if ∥ T x ∥ L 2 ≤ C ∥ x (0) ∥ 2) PIE FE stable if ∥ x ∥ L 2 ≤ C ∥ x (0) ∥ 3) PDE FE stable if ∥ T x ∥ L 2 ≤ C ∥ T x (0) ∥ 4) PDE to PIE FE stable if ∥ x ∥ L 2 ≤ C ∥ T x (0) ∥ for some C > 0 and an y x ( t ) which satisfies T ˙ x ( t ) = A x ( t ) . L yapunov Conditions for FE stability Unlike exponential stability , negati vity of the deriv ative is critical in FE stability . Hence, we do not use L yapunov negativity in Lemma 5. Lemma 5 (Conditions for F inite Ener gy Stability): Suppose V is positive semidefinite. Then T ˙ x = A x is • FE PIE to PDE stable if V is PIE bounded and PDE negati ve. • FE PIE stable if V is PIE bounded and PIE ne gati ve. • FE PDE stable if V is PDE bounded and PDE negativ e. • FE PDE to PIE stable if V is PDE bounded and PIE negati ve. Pr oof: In each case, we use V ( x ) ≥ 0 and inte grate the L yapunov negati vity condition. Specifically , for the PDE negati vity property , we integrate ˙ V ( x ( t )) ≤ − α ∥ T x ( t ) ∥ 2 to show that for an y T , V ( x ( T )) ≥ 0 and hence V ( x ( T )) − V ( x (0)) ≤ − α Z T 0 ∥ T x ( t ) ∥ 2 dt ⇒ Z T 0 ∥ T x ( t ) ∥ 2 dt ≤ 1 α V ( x (0)) . W e may then take the limit as T → ∞ to sho w that α ∥ T x ∥ 2 L 2 ≤ V ( x (0)) . For PIE negati vity , we similarly hav e α ∥ x ∥ 2 L 2 ≤ V ( x (0)) . Now , statement 1) follows since α ∥ T x ∥ 2 L 2 ≤ V ( x (0)) ≤ C ∥ x (0) ∥ 2 . Statement 2) from α ∥ x ∥ 2 L 2 ≤ V ( x (0)) ≤ C ∥ x (0) ∥ 2 . Statement 3) from α ∥ T x ∥ 2 L 2 ≤ V ( x (0)) ≤ C ∥ T x (0) ∥ 2 . Statement 4) from α ∥ x ∥ 2 L 2 ≤ V ( x (0)) ≤ C ∥ T x (0) ∥ 2 . Note 2: There is a significant question of whether FE stability implies asymptotic stability . Suppose we have FE PIE stable or FE PDE to PIE stable. Then x ∈ L 2 and hence ∥ T ˙ x ∥ L 2 = ∥ A x ∥ L 2 ≤ ∥ A ∥∥ x ∥ L 2 Hence T ˙ x ∈ L 2 , which implies lim t →∞ ∥ T x ( t ) ∥ → 0 . Unfortunately , ho wev er, this only holds for FE PIE stable and FE PDE to PIE stable. V . S U FFI C I E N T C O N D I T I O N S F O R N OT I O N S O F S TA B I L I T Y In this section, we briefly provide a list of relati vely obvious conditions for the various stability notions in terms of operator inequality constraints on the PI operators T and A , as well as some operator v ariable, P ∈ Π 3 or Q ∈ Π 3 . Here the variable label P will be used when the v ariable is constrained to be self-adjoint and Q otherwise. The operator inequality constraints are all interpreted as positivity on L 2 and may be tested using discretization or analytically by finding a Mercer representation of the kernel. Alterna- tiv ely , operator inequalities may be enforced numerically using semidefinite programming through software such as PIETOOLS [9]. Lemma 6 (LPIs for L yapunov Bounds): Gi ven T , if V ( x ) = ⟨ x , P x ⟩ , then 1) positiv e semidefinite is equi valent to P ≥ 0 2) PDE positivity is equi valent to P ≥ ϵ T ∗ T 3) PIE positivity is equi valent to P ≥ ϵI 4) PDE bounded is equiv alent to P ≤ C T ∗ T 5) PIE bounded is equiv alent to P ≤ C I for some ϵ, C > 0 . If ˙ V ( x ) = ⟨ x , D x ⟩ , then 1) PDE negati ve is equiv alent to D ≤ − α T ∗ T 2) PIE negati ve is equiv alent to D ≤ − αI 3) L yapunov neg ativ e is equi valent to D ≤ − α P for some α > 0 . Note that PIE bounded in satisfied for any P ∈ Π 3 . Every sufficient condition proposed in this section relies on one of the follo wing two canonical candidate L yapunov functions. Specifically , giv en T , we hav e V 1 ( x ) := ⟨ T x , PT x ⟩ and V 2 ( x ) := ⟨ x , QT x ⟩ where V 1 and V 2 are parameterized by PI operators P ∈ Π 3 and Q ∈ Π 3 , respecti vely . Here we constrain P = P ∗ and QT = T ∗ Q ∗ . Properties of V 1 : First consider V 1 ( x ) , which is perhaps the most obvious form of L yapunov function, being defined solely in terms of the PDE state, T x . Applying Lem. 6 with P 7→ T ∗ PT , V 1 is PDE positive if P ≥ ϵI . Howe ver , it is not usually possible for V 1 to be PIE positiv e, since this would imply T ∗ PT ≥ ϵI . If we recall that typically T ∈ Π 2 (or at least some part thereof), this implies that T ∗ PT ∈ Π 2 and hence cannot be coerciv e. Next, we note that V 1 is both PDE and PIE bounded, since T ∗ PT ≤ ∥ P ∥ T ∗ T implies PDE bounded and PDE bounded implies PIE bounded. For T ˙ x = A x , the deriv ative of V 1 is ˙ V 1 ( x ) := ⟨ T x , PA x ⟩ + ⟨ PA x , T x ⟩ = ⟨ x , ( T ∗ PA + A ∗ PT ) x ⟩ . Thus ˙ V 1 is PDE negati ve if T ∗ PA + A ∗ PT ≤ − ϵ T ∗ T . Howe ver , as for PIE positi vity , ˙ V is unlikely to be PIE negati ve since this would require T ∗ PA + A ∗ PT ≤ − ϵI which is not possible unless T , A ∈ Π 3 . Properties of V 2 : Consider now V 2 = ⟨ x , QT x ⟩ where QT = ( QT ) ∗ . The difference between V 1 and V 2 is that V 2 is defined in terms of both the PDE state ( T x ) and PIE state ( x ). For example, in the heat equation, for Q = − I we would hav e V = −⟨ u , u ss ⟩ which reduces to something like V = ⟨ u s , u s ⟩ , depending on boundary conditions. Applying Lem. 6 with P 7→ QT , V 2 is PDE positiv e if QT ≥ ϵ T ∗ T . As with V 1 , PIE positivity is not possible since it implies that QT ∈ Π 2 . V 2 is always PIE bounded and is PDE bounded if QT ≤ C T ∗ T . For T ˙ x = A x , the deriv ativ e of V 2 is ˙ V 2 ( x ) := ⟨ x , QA x ⟩ + ⟨ QA x , x ⟩ = ⟨ x , ( QA + A ∗ Q ∗ ) x ⟩ . Thus ˙ V 2 is PDE negativ e if QA + A ∗ Q ∗ ≤ − ϵ T ∗ T and PIE negati ve if QA + A ∗ Q ∗ ≤ − αI . Unlike ˙ V 1 , it is possible for ˙ V 2 to be PIE negati ve. Finally , note that V 1 is a special case of V 2 with Q = T ∗ P . A. Operator Inequalities for L yapunov Stability In the following corollary , we combine V 1 and V 2 with Lemmas 3 and 6 to obtain the following operator inequalities for each category of L yapunov stability . Cor ollary 7: Gi ven T , A , the PIE T ˙ x = A x is 1) PIE to PDE stable if QT = ( QT ) ∗ ≥ ϵ T ∗ T and QA + A ∗ Q ∗ ≤ 0 . 2) PIE stable if QT = ( QT ) ∗ ≥ ϵI and QA + A ∗ Q ∗ ≤ 0 . 3) PDE stable if P ≥ ϵI , T ∗ PA + A ∗ PT ≤ 0 . 4) PDE to PIE stable if QT = ( QT ) ∗ ≥ ϵI , QT ≤ C T ∗ T , and QA + A ∗ Q ∗ ≤ 0 . for some Q or P , ϵ, C > 0 . Example 1 (PIE to PDE Stability of the W ave Equation): Let us no w reconsider the wav e equation which we recall is not PDE stable. Specifically , the wa ve equation u tt = u xx with u ( t, 0) = u ( t, 1) = 0 admits a PIE representation of the form T z }| {  I 0 0 T 0   ˙ u 1 ˙ u 2  = A z }| {  0 I I 0   u 1 u 2  . where u 1 = ˙ u and u 2 = u xx and where ( T 0 x )( s ) := Z 1 0 G T ( s, θ ) x ( θ ) dθ G T ( s, θ ) = ( − θ θ < s − s s < θ . T o apply the stability conditions of Lemma 3, we observe that T 0 = − R ∗ R for ( R x )( s ) := Z 1 0 G R ( s, θ ) x ( θ ) dθ G R ( s, θ ) = ( 0 θ < s − 1 s < θ . Thus if we choose Q =  I 0 0 − I  , we find that T ∗ 0 T 0 = R ∗ RR ∗ R ≤ ∥ RR ∗ ∥ R ∗ R = −∥ T 0 ∥ T 0 and QT = T ∗ Q ∗ =  I 0 0 − T 0  ≥ min  1 , 1 ∥ T 0 ∥   I 0 0 T ∗ 0 T 0  = min  1 , 1 ∥ T 0 ∥  T ∗ T . Furthermore, QA + A ∗ Q ∗ =  I 0 0 − I   0 I I 0  +  0 I I 0   I 0 0 − I  = 0 . Hence we may use Lemma 7 to conclude PIE to PDE stability of the wave equation. B. Operator Inequalities for Exponential Stability W e combine V 1 and V 2 with Lemmas 4 and 6 to obtain the following operator inequalities for each category of exponential stability . Lemma 8 (LPIs for Exp. Stability): Giv en T , A , the PIE T ˙ x = A x is 1) Exp. PIE to PDE stable if QT = ( QT ) ∗ ≥ ϵ T ∗ T , QA + A ∗ Q ∗ ≤ − α QT 2) Exp. PIE stable if QT = ( QT ) ∗ ≥ ϵI and QA + A ∗ Q ∗ ≤ − α QT 3) Exp. PDE stable if P ≥ ϵI , T ∗ PA + A ∗ PT ≤ − α T ∗ PT 4) Exp. PDE to PIE stable if QT = ( QT ) ∗ ≥ ϵI , QT ≤ C T ∗ T , and QA + A ∗ Q ∗ ≤ − α QT for some Q or P and ϵ, C, α > 0 . If we wish to avoid bilinearity in decision variables α and Q or P , we may replace the L yapunov negati vity conditions in Lem. 4 with PIE or PDE negativity . Recalling the relationship between L yapunov , PDE, and PIE negati vity , we hav e the following. Lemma 9 (Alternative LPIs for Exp. stability): Giv en T , A , the PIE T ˙ x = A x is 1) Exp. PIE to PDE stable if QT = ( QT ) ∗ ≥ ϵ T ∗ T , QA + A ∗ Q ∗ ≤ − αI 2) Exp. PIE stable if QT = ( QT ) ∗ = QT ≥ ϵI and QA + A ∗ Q ∗ ≤ − αI 3) Exp. PDE stable if P ≥ ϵI , T ∗ PA + A ∗ PT ≤ − α T ∗ T 4) Exp. PDE to PIE stable if QT = ( QT ) ∗ ≥ ϵI , QT ≤ C T ∗ T , and QA + A ∗ Q ∗ ≤ − αI for some Q or P and ϵ, C, α > 0 . Pr oof: Statements 1) and 2) replace L yapunov negati ve with PIE negati ve and bounded. Statements 3) and 4) replace L yapunov neg ativ e with PDE negati ve and bounded. Example 2 (The Heat Equation is Exp. PDE stable): The heat equation u t = u ss + λ u with Dirichlet boundary conditions admits a PIE representation T ˙ x = A x = ( I + λ T ) x where ( T x )( s ) = Z s 0 θ ( s − 1) x ( θ ) dθ + Z 1 s s ( θ − 1) x ( θ ) dθ and T = T ∗ = − M ∗ M where ( M x )( s ) = Z s 0 θ x ( θ ) dθ + Z 1 s ( θ − 1) x ( θ ) dθ . Now , − M ∗ M ≤ − π 2 T ∗ T . Hence for P = I , we have that T ∗ PA + A ∗ PT = − 2( M ∗ M − λ T ∗ T ) ≤ − 2( π 2 − λ ) T ∗ T . Hence by Lemma 8 we hav e that the heat equation is exponentially PDE stable for any λ < π 2 . C. Operator Inequalities for F inite Ener gy Stability Finally , we combine V 1 and V 2 with Lemmas 5 and 6 to obtain the following operator inequalities for each category of finite energy stability . Lemma 10: Gi ven T , A , the PIE T ˙ x = A x is 1) FE PIE to PDE stable if QT = ( QT ) ∗ ≥ 0 , QA + A ∗ Q ∗ ≤ − α T ∗ T . 2) FE PIE stable if QT = ( QT ) ∗ ≥ 0 and QA + A ∗ Q ∗ ≤ − αI . 3) FE PDE stable if P ≥ 0 , T ∗ PA + A ∗ PT ≤ − α T ∗ T . 4) FE PDE to PIE stable if QT = ( QT ) ∗ ≥ 0 , QT ≤ C T ∗ T , and QA + A ∗ Q ∗ ≤ − αI . for some Q or P and ϵ, C, α > 0 . Note that unlike L yapunov or exponential PIE stability , FE PIE stability is often possible to v erify . V I . S U M M A RY T able I summarizes the weakest L yapunov conditions required for each notion of stability . V I I . C O N C L U S I O N W e have proposed a hierarchical classification of L ya- punov , exponential, and finite-energy stability notions in terms of the fundamental state, x , which is related to the PDE state as x = D α u and u = T x and which satisfies T ˙ x = A x for some PI operators, T , A . This framework allo ws us to unify man y existing L yapunov stability conditions which may or may not include partial deriv ativ es of the PDE state in L yapunov function lower bounds, upper bounds, and ne gati vity conditions. Specifically , L yapnunov conditions which establish stability , but not PDE stability will typically imply PIE to PDE stability . This framew ork allo ws for classification of PDEs such as the wave and beam equations, which are not PDE stable. Furthermore, the weakest possible L yapunov and operator L 2 -inequality conditions are pro vided for each notion of stability . Such conditions may be verified analytically or using numerical software such as PIETOOLS. R E F E R E N C E S [1] T . Ha-Duong and P . Joly , “On the stability analysis of boundary con- ditions for the wave equation by energy methods. I. the homogeneous case, ” mathematics of computation , vol. 62, no. 206, pp. 539–563, 1994. [2] M. A. Beck, “T opics in stability theory for partial dif ferential equa- tions, ” Ph.D. dissertation, Boston University , 2006. [3] A. Mironchenko and F . Wirth, “Non-coercive L yapunov functions for infinite-dimensional systems, ” Journal of Differ ential Equations , vol. 266, no. 11, pp. 7038–7072, 2019. [4] B. Jacob, A. Mironchenko, J. Partington, and F . Wirth, “Remarks on input-to-state stability and non-coercive L yapunov functions, ” in IEEE Confer ence on Decision and Contr ol , 2018, pp. 4803–4808. [5] ——, “Noncoerciv e lyapunov functions for input-to-state stability of infinite-dimensional systems, ” SIAM Journal on Contr ol and Optimiza- tion , vol. 58, no. 5, pp. 2952–2978, 2020. [6] M. Peet., “ A partial integral equation representation of coupled linear PDEs and scalable stability analysis using LMIs, ” Automatica , vol. 125, March 2021. [7] D. Jagt and M. Peet, “ A state-space representation of linear multiv ariate PDEs and stability analysis using SDP, ” Automatica , 2025, submitted, [8] S. Shi vakumar , A. Das, S. W eiland, and M. Peet, “Extension of the partial integral equation representation to GPDE input-output systems, ” IEEE Tr ansactions on Automatic Control , vol. 70, no. 5, 2024. [9] S. Shiv akumar, D. Jagt, D. Braghini, A. Das, Y . Peet, and M. Peet, PIETOOLS 2025: User Manual , arXiv . Positivity Upper Bound Negati vity Notion PIE PDE PDE PIE PIE PDE SD PIE2PDE X X X PIE X X X PDE X X X PDE2PIE X X X Exp PIE2PDE X X X Exp PIE2PDE X X X Exp PIE X X X Exp PIE X X X Exp PDE X X X Exp PDE2PIE X X X FE PIE2PDE X X FE PIE X X FE PDE X X FE PDE2PIE X X T ABLE I: The weakest L yapunov conditions which imply each given notion of stability (2 v ariants for PIE2PDE and PIE exp. stability). Cate gories of positivity , bound, and negati vity are ordered from strongest to weakest.