ISS Estimates in the Spatial Sup-Norm for Nonlinear 1-D Parabolic PDEs

This paper provides novel Input-to-State Stability (ISS)-style maximum principle estimates for classical solutions of highly nonlinear 1-D parabolic Partial Differential Equations (PDEs). The derivation of the ISS-style maximum principle estimates is performed by using an ISS Lyapunov Functional for the sup norm. The estimates provide fading memory ISS estimates in the sup norm of the state with respect to distributed and boundary inputs. The obtained results can handle parabolic PDEs with nonlinear and non-local in-domain terms/boundary conditions. Three illustrative examples show the efficiency of the proposed methodology for the derivation of ISS estimates in the sup norm of the state.


Introduction
Rapid progress in the development of Input-to-State Stability (ISS) theory for systems modeled by Partial Differential Equations (PDEs) took place during the last decade (see for instance [2,6,7,8,11,13,15,19,20,21,25,26]). Researchers dealt with the two major problems that arise in the study of PDEs with inputs and do not appear in the study of finite-dimensional systems or delay systems: (i) the selection of the state norm (functional norms are not equivalent), and (ii) the presence of boundary inputs (the boundary inputs enter through unbounded operators). Many methodologies have been used for the derivation of ISS estimates (Lyapunov functionals, spectral methods, semigroup methods, etc.). The development of ISS theory for PDEs has produced novel results which can be used for various purposes in control theory and practice. ISS theory for PDEs has been used for the stability analysis of coupled PDEs (see [2,15]), for feedback design purposes in linear and semilinear PDEs (see [12,14,24]) and for observer design purposes (see [16]) for various infinite-dimensional systems.
Although the sup norm is the most useful norm (because it can provide point-wise estimates), there are few ISS estimates in the literature for the sup norm of the solution of parabolic PDEs. The scarcity of ISS estimates in the state sup norm is justified by the absence of ISS Lyapunov functionals which are related to the sup norm. ISS estimates in the sup norm were provided in the work [11], where numerical approximations of the solution of a linear 1-D parabolic PDE were exploited. The methodology was later generalized in [15] by introducing the notion of an ISS Lyapunov functional under discretization. Input-to-Output Stability (IOS) estimates for the state sup norm were provided for certain semilinear parabolic PDEs in [26]. Indeed, in [26] the state sup norm was estimated by using the 1 (0,1) H norm of the initial condition and consequently, the provided estimates were IOS estimates with output map being the state in the space (0,1) L  while the state space was the Sobolev space 1 (0,1) H . ISS estimates of the 1 (0,1) H norm of the state for certain linear 1-D parabolic PDEs were also provided in [13]. These estimates can be used for the derivation of estimates of the sup norm by using auxiliary tools (e.g., Agmon's inequality).
The present work provides ISS estimates in the sup norm for highly nonlinear and non-local 1-D parabolic PDEs by using a completely different methodology than the methodology used in [11,15]. Instead of exploiting numerical approximations of the solution of the PDE problem, here we construct an ISS Lyapunov Functional for the sup norm. To that purpose, we consider classical solutions of the 1-D parabolic PDE (as in [11,15] but less regular solutions than the solution notion in [11,15]). The constructed ISS Lyapunov functional is a coercive Lyapunov functional (i.e., is bounded from above and below by K  functions of the sup norm of the state). It should be noticed that non-coercive Lyapunov functionals may be used for the proof of the ISS property (see [8]). Using the coercive ISS Lyapunov functional, we are able to get more general results than the results in [11,15], which can deal with highly nonlinear (and non-local) parabolic PDEs and highly nonlinear (and non-local) boundary conditions. More specifically, we obtain ISS-style maximum principle estimates which (like standard maximum principles; see [3,9,22]) provide bounds for the sup norm of the state that depend only on the sup norm of the initial condition, the source terms in the PDE and the boundary values of the solution (and its spatial derivative). However, unlike standard maximum principles, the obtained estimates take into account the exponential decay of the effect of the initial condition and the fact that recent input values have stronger effect on the current value of the solution than past input values (fading memory effect). The ISS-style maximum principle estimates provide in a direct way fading memory ISS estimates when disturbance inputs appear in the boundary conditions. The proof methodology is simpler and less lengthy than the methodology used in [11,15].
The structure of the paper is as follows. Section 2 contains the statements of the main results of the paper and provides certain remarks. The main results are used in Section 3 in three (carefully chosen) illustrative examples, which show how easily the main results can be applied to highly nonlinear parabolic PDEs. The proofs of the main results are provided in Section 4, where some interesting auxiliary results are also stated and proved. Section 5 gives the concluding remarks of the present work.
Notation. Throughout this paper, we adopt the following notation.  Let n U  be a set with non-empty interior and let    be a set. By 0 ( ; ) CU , we denote the class of continuous mappings on U , which take values in  . By

ISS in the Spatial Sup-Norm
Our first main result provides an estimate of a weighted sup norm of the solution of a PDE with space and time-varying coefficients. The estimate depends only on the initial condition, the distributed perturbation term that may be present in the PDE and the boundary values of the solution.
Theorem 2.1: exists, is a continuous function on (0, ) [0,1] T  and satisfies for almost all (0, ) tT  and for all (0,1) 2) and that there exist a constant 0   and a positive function Then the following estimate holds for all [ [3,9,22]) provides a bound for the sup norm of the state that depends only on the sup norm of the initial condition, the source terms in the PDE ( f ) and the boundary values of the solution. However, unlike ordinary maximum principles, estimate (2.4) takes into account the exponential decay of the effect of the initial condition and the fact that past input values have less effect on the current value of the solution than more recent input values (fading memory effect the boundary values of the state ( , 0), ( ,1) u t u t and the distributed perturbation term f that appears in the PDE (2.1). An ISS Lyapunov functional for the PDE system (2.1) satisfies the following properties (see also [15] Chapter 1 for a slightly more demanding notion of an ISS Lyapunov functional for (2.1)): ".  [3,9,22]). Theorem 2.1 holds for classical solutions of the PDE (2.1). Existence and uniqueness results for classical solutions of (nonlinear) parabolic PDEs are given in [3,17].
A disadvantage of Theorem 2.1 is the fact that Theorem 2.1 can provide ISS estimates only when boundary disturbances appear in Dirichlet boundary conditions. However, boundary disturbances can appear in other types of boundary conditions as well (e.g., Neumann or Robin boundary conditions). Our next main result provides an estimate of a weighted sup norm of the solution of the PDE (2.1) that depends on the initial condition, the distributed perturbation term that may be present in the PDE and the boundary values of the solution and its spatial derivative.

Theorem 2.2:
Let exists and is a continuous function on (0, ) [0,1] T  , the derivatives It should be noted that estimate (2.6) is a more accurate ISS-style maximum principle estimate than However, there is a price to pay for obtaining the more accurate estimate (2.6): the solution has to be differentiable with respect to x up to the boundary of the domain. This additional regularity requirement does not appear in Theorem 2.1. This additional regularity requirement is justified by the fact that Theorem 2.2 is a tool for handling boundary conditions which involve the spatial derivative at the boundary of the domain.
The functions 01 , : (0, ] (0, ) g g T   , 01 , : (0, ] [1, ) k k T   are arbitrary and can be selected in an appropriate way in order to handle even nonlinear boundary conditions (see Example 3.2 in next section). However, when standard Neumann or Robin boundary conditions hold at the boundary then we obtain as a corollary of Theorem 2.2 the following result.

exists and is a continuous function on
14) The ISS-style maximum principle estimate (2.6), with 01 , rr defined by (2.12) or (2.13) or (2.14), provides in a direct way a fading memory ISS estimate when we have the boundary conditions Conditions (2.9), (2.10), (2.11) are not new: they have also appeared in [11,15], where a completely different proof methodology was applied (the solution was approximated by a numerical scheme). However, here we do not follow the proof methodology in [11,15] and we obtain Corollary 2.3 under weaker requirements for the solution.

Illustrative Examples
In this section we present three examples that illustrate the use of the main results for the derivation of ISS estimates for nonlinear 1-D parabolic PDEs. The following example illustrates how Theorem 2.2 can be used for the derivation of ISS estimates for parabolic PDEs with nonlinear and non-local boundary conditions. Moreover, the example shows that we can study parabolic PDEs for which the diffusion coefficient is a non-local functional of the state. This situation is important because it arises in many phenomena related to turbulent flows (see [4,5]) as well as in the work of O. A. Ladyzhenskaya (see [18]) on the modification of the Navier-Stokes equations and in feedback control of fluid flows (see [1]).

Example 3.2 (Heat Equation with Nonlinear and Non-Local Boundary Conditions):
Consider the following nonlinear and non-local 1-D parabolic PDE: The final example shows that the main results of the present work can be used for the study of realistic problems where the physical properties of a material are nonlinear functions of thermodynamic variables (such as temperature or pressure).

Example 3.3 (The "real" heat equation):
Thermal conductivity k and volumetric heat capacity c are quantities that depend on the temperature u of a solid material. Consequently, by exploiting the conservation of energy and Fourier's law, we obtain the following equation for a 1-D spatial domain: Equation (3.12) is important for certain solids when the temperature variation is large. Therefore, we are led to the study of the PDE problem  semiconductors, crystalline quartz and crystalline ceramics, used in the manufacture of quartz clocks and various kinds of electronic devices, including diodes, transistors, and integrated circuits, are known to have a thermal conductivity that scales as 1/u (see [23]). With further calculations, which would need to be mindful of the absolute zero and melting point limits on the temperature, one could pursue deriving a growth rate on the ISS gain

Proofs of Main Results
For the proofs of the main results of the paper we need some auxiliary results. The first auxiliary result provides a sufficient condition that guarantees that the sup norm is an absolutely continuous function of time.

  
Similarly as above, we get:   Consequently, the two above inequalities guarantee that the following inequality holds for all 12 ,   is arbitrary, the above inequalities allow us to conclude that

Equality (4.1) is a direct consequence of equation (4.3).
The second auxiliary result is a technical lemma that provides a formula for a specific limit. This specific limit was encountered in the statement of Lemma 4.1 and consequently, the formula that the following lemma provides, plays a crucial role.
for all 0 h  sufficiently small. The last equality in (4.7) The above inequality together with the fact that consequence of the fact that xI   ) implies that 0 p  ; a contradiction with (9). Similarly, we obtain a contradiction when ( ) 0 n ux  for sufficiently large n . The proof is complete.
We are now ready to derive estimates for the solution of the PDE (2.1) under assumption (2.2). The third auxiliary result provides an estimate under an additional assumption for the coefficient of the reaction term ( , ) c t x .

4.3:
Let Implication (4.17) and Lemma 2.14 on page 82 in [10] imply that the following estimate holds for all 12 [ , ] t t t  :

Concluding Remarks
The present paper provided tools for the derivation of ISS estimates in the sup norm of the state. More specifically, we provided novel ISS-style maximum principle estimates which are valid for classical solutions of parabolic PDEs. The obtained results can be applied to highly nonlinear 1-D parabolic PDEs, as illustrated by three illustrative examples. However, the main results have some disadvantages. The main disadvantages of the main results of the paper are: 1) the fact that they can only be applied to PDEs for which classical solutions can be proved to exist (at least locally; see [3,17]). In some cases, we are in a position to prove the existence of a continuous solution (so that the sup norm estimates make sense) which satisfies the PDE in a weak sense (see for example [25]). For such cases, Theorem 2.1 and Theorem 2.2 cannot be applied. Therefore, there is a need to extend the results for weaker notions of solutions. This is going to be the topic of future research.
2) the fact that they can only be applied to parabolic PDEs with one spatial dimension. The extension to parabolic PDEs with n  dimensional domains is going to be another topic of future research.