Stability Analysis of Parabolic Linear PDEs with Two Spatial Dimensions Using Lyapunov Method and SOS

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1 Staility Analysis of Paraolic Linear PDEs with Two Spatial Dimensions Using Lyapunov Metho an SOS Evgeny Meyer an Matthew M. Peet Astract In this paper, we aress staility of paraolic linear Partial Differential Equations PDEs. We consier PDEs with two spatial variales an spatially epenent polynomial coefficients. We parameterize a class of Lyapunov functionals an their time erivatives y polynomials an express staility as optimization over polynomials. We use Sum-of-Squares an Positivstellensatz results to numerically search for a solution to the optimization over polynomials. We also show that our algorithm can e use to estimate the rate of ecay of the solution to PDE in the L norm. Finally, we valiate the technique y applying our conitions to the D iological KISS PDE moel of population growth an an aitional example. I. INTRODUCTION Staility analysis an controller esign for Partial Differential Equations PDEs is an active area of research [], []. One approach to staility analysis of PDEs is to approximate the PDEs with Orinary Differential Equations ODEs using, e.g. Galerkin s metho or finite ifference, an then apply finite-imensional optimal control methos, e.g. [3] [6]. In this paper we consier staility analysis without iscretization. Specifically, we use Sum-of-Squares SOS optimization to construct Lyapunov functionals for paraolic PDEs with two spatial variales an spatially epenent coefficients. It is well-known that existence of a Lyapunov function for a system of ODEs or PDEs is a sufficient conition for staility. For example, [7] uses a Lyapunov approach an Linear Operator Inequalities LOIs to provie sufficient conitions for exponential staility of a controlle heat an elaye wave equations. Another metho ase on Lyapunov an semigroup theories was applie in [8] for analysis of wave an eam PDEs with constant coefficients an elaye ounary control. In [9] a Lyapunov ase analysis of semilinear iffusion equations with elays gave staility conitions in terms of Linear Matrix Inequalities LMIs. Extensive examples of applying the ackstepping metho to the ounary control of PDEs can e foun in [] [4]. Briefly speaking, ackstepping uses a Volterra operator to search for an invertile mapping from the original PDE to a chosen target PDE, known to e stale. In orer to fin such mapping, one has to solve analytically or numerically a PDE for Volterra operator s kernel. If the mapping is foun, This work was supporte y the National Science Founation uner Grants No. 385 an 366. Evgeny Meyer is a Ph.D stuent with the School for Engineering of Matter, Transport an Energy, Arizona State University, Tempe, AZ, 858 USA emeyer@asu.eu Matthew M. Peet is an assistant professor with the School for Engineering of Matter, Transport an Energy, Arizona State University, Tempe, AZ, 858 USA mpeet@asu.eu it provies the ounary control law. Applications for twoimensional cases were iscusse in [5] an [6]. However, ackstepping requires us to guess on the target PDE an solve the PDE for kernel, which may e a challenging task for PDEs with two spatial variales an spatially epenent coefficients. Moreover, ackstepping cannot e use for staility analysis in the asence of a control input. Note that SOS has een previously applie in [7] an [8] to fin Lyapunov functionals for D paraolic PDE. Input- Output analysis of PDE systems with SOS implementation is iscusse in [9]. Examples of using SOS in controller esign for one-imensional PDEs can e foun in [] []. The main contriution of this paper is to provie an algorithm for staility analysis of D paraolic PDEs with spatially varying coefficients. We search for polynomial s which efines a Lyapunov functional of the form V ut, = sxu x where u is the solution of the PDE, t represents time, x an are the spatial variale an omain. We introuce staility conitions in the form of parameter epenent LMIs, which certify Lyapunov inequalities, i.e. V ut, > an [V ut, ] < for all t >. There exist several algorithms for optimization over polynomials which can e applie to in orer to fin V. See e.g., [4] for a survey on algorithms for optimization of polynomials. In this paper, we apply SOS an the Positivstellensatz results to fin a solution to parameter epenent LMIs. For etails on Positivstellensatz see [3] or [5]. We use the MATLAB toolox SOSTOOLS see [6] for etails on SOSTOOLS an SeDuMi, a well-known semiefinite programming solver, to solve the SOS optimization prolem. The paper is organize as follows. In Section we specify the notation an provie some ackgroun. Lyapunov staility conitions for paraolic PDEs are given in Section 3. We emonstrate the propose metho in Section 4 an present SOS optimization prolems in Section 5. Finally, we iscuss numerical implementations in Section 6 an conclue the paper in Section 7 with ieas aout future research steps. A. Notation II. PRELIMINARIES N is the set of natural numers an N := N {}. R n an S n are the n-imensional Eucliean space an space of n n real symmetric matrices. For x R n, let x T enote transpose x an x i R is the i-th component of x. is

2 a norm on R n, efine as x := n i= x i. For X S n, X means that X is negative semiefinite. The symol will enote the symmetric elements of a symmetric matrix. For R n an f : R let fx stan for fx,..., x n an fx x represent an integral of f over with x := x x...x n. Let N n := {α R n : α i N }. A vector α N n is calle multi-inex. For l N efine the set Q n l := {α N n : α l}. For α N n, x R n an g : R n R partial erivative D α [gx] := α n x α [gx] = αi [gx]. 3 x αi i i= Note that [gx] = gx for any i {,..., n}. We will x i also use classical notation, u x x := x [ x [u]]. If for a function f : R an some α N n erivative D α [fx] exists for all x, there exists g : R such that gx = D α [fx] for all x. For revity D α [f] := g. Further we use W,. It is one of Banach spaces W k,p which enote Soolev spaces of functions u : R with D α [u] L p for all α Q n k, where Qn k is efine as in an norm u k,p := D α [u] Lp, α k where L p stans for the space of Leesgue-measurale functions g : R with norm, for p N /p g Lp := gs p s an g L := sup s gs. It is known that for continuous functions u : [, W, an V : W, R their composition V u : [, R is also continuous an the upper right-han erivative D t + V ut is efine y D + [V ut] := lim sup h + V ut + h V ut. h Note that if v : [, R is ifferentiale at t, then D + [vt] = [vt]. B. Polynomials For a multi-inex α N n an x R n, let x α := n i= xαi i = x α xα... xα n n. Then x α is a monomial of egree α N. A polynomial is a finite linear comination of monomials px := α p αx α, where the summation is applie over a given finite set of multi-inexes α an p α R enotes the corresponing coefficient. Let R[x] stan for polynomial ring in n variales x,..., x n with coefficients in R. The egree of a polynomial p R[x] is the largest egree among all monomials, an is enote y egp N. A polynomial p R[x] is calle Sum of Squares SOS, if there is a finite numer of polynomials z i R[x] such that for all x R n, px = i z ix. We enote the set of sum of square polynomials in variales x y [x]. If p [x], then px for all x R n. Polynomial matrices an SOS matrices are efine in a similar manner See, e.g. [7]. We use [Sx] to enote the set of real symmetric SOS polynomial matrices. If M [S m x] then for all x R n, Mx S m an Mx. C. Comparison Lemma Recall the comparison principle from, e.g. [8]. Lemma : Consier the scalar ifferential equation [ut] = ft, ut, ut = u, where f is continuous in t an locally Lipschitz in x, for all t an all x J R. Let [t, T T coul e infinity e the maximal interval of existence of the solution u, an suppose ut J for all t [t, T. Let v e a continuous function whose upper right-han erivative D + [vt] satisfies D + [vt] ft, vt, vt u with vt J for all t [t, T. Then, vt ut for all t [t, T. III. LYAPUNOV STABILITY FOR PARABOLIC PDE First consier the following general form of paraolic PDEs. For all t, an x R n, u t = F t, x, D α [u],..., D αk [u], 4 where u : [, R an for i =,..., k, D αi [u] are partial erivatives as enote in 3. Assume that solutions to 4 exist, are unique an epen continuously on initial conitions. This implies that the function is continuously ifferentiale in t an for each t [,, ut, W,. Theorem : Let there exist continuous V : L R, l, m N an, a > such that a v l L V v v m L, 5 for all v L. Furthermore, suppose that there exists c such that for all t the upper right-han erivative D + [V ut, ] c ut, m L, 6 where u satisfies 4. Then { ut, L l u, m/l L a exp c } l t for all t. Proof: Let conitions of Theorem e satisfie. Since W, L, from 5 it follows that for each t a ut, l L V ut, ut, m L. 7 Diviing oth sies of the secon inequality in 7 y results in V ut, ut, m L. 8 After multiplying oth sies of 8 y c, we have c ut, m L c V ut,. 9 From 6 an 9 it follows that D + [V ut, ] c V ut,. To use the comparison principle, consier the ODE [ϕt] = c ϕt, ϕ = V u,,

3 where t, an function ϕ : [, R is continuous. The well-known solution for is ϕt = V u, exp t for all t. Applying Lemma for an results in V ut, V u, exp t for all t. Sustituting t = in the secon inequality of 7 implies V u, u, m L. 3 Comining the first inequality of 7 with 3 an gives a ut, l L V ut, V u, exp t u, m L exp t. 4 Diviing 4 y a an taking the l th root results in { ut, L l u, m/l L a exp c } l t for all t. IV. STABILITY TEST EXPRESSED AS OPTIMIZATION OVER POLYNOMIALS In this paper, we focus on PDEs of the following form. For all t > an x :=,, u t = axu x x + xu x x + cxu xx + xu x + exu x + fxu, 5 where a,, c,, e, f R[x]. Assume that solution to 5 exists, is unique an epens continuously on initial conitions. For each t suppose ut, W,. Furthermore, let u satisfy zero Dirichlet ounary conitions, i.e. ut,, x = ut,, x = ut, x, = ut, x, = 6 for all x, x [, ]. Define V : L R as V v := sxvx x, 7 where s R[x]. Using ut, for v in 7 an ifferentiating the result with respect to t gives [V ut, ] = [ ] sxu x = sxuu t x. 8 Sustituting for u t from 5 into 8 implies [V ut, ] = sxu axu xx + xu x x + cxu x x + xu x + exu x + fxu x = I t + I t + I 3 t + I 4 t + I 5 t, 9 where I t := I t := I 3 t := I 4 t := I 5 t := sxuaxu x x x, sxuxu x x x, sxuxu xx x, sxucxu x x x, sxu xu x + exu x + fxu x. Note that, ase on section 5..3 of [9], we have u x x = u x x for all x. We use property to efine I an I 3. Alternatively, I 5 can e formulate as I 5 t = U T Z 5 xu x, where for all x U T := [u u x u x ] an sxfx sxx sxex Z 5 x :=. Using integration y parts an ounary conitions 6, I can e rewritten as follows. I t = sxuax [u x ] x x = sxuaxu x x = x = u x [sxuax] x x x = u x u [sxax] x + sxaxu x x = U T Z xu x, 3 where for all x x [sxax] Z x := sxax. Following steps of 3 for I, I 3 an I 4, we get I t = sxux [u x ] x x = sxuxu x x= x = u x [sxux] x x x = U T Z xu x,

4 I 3 t = = = I 4 t = = sxux [u x ] x x sxuxu x x = x = u x x [sxux]x x U T Z 3 xu x, sxucx [u x ] x x U T Z 4 xu x, 4 where U is efine as in an for all x x [sxx] Z x := sxx, x [sxx] Z 3 x := sxx, x [sxcx] Z 4 x :=. sxcx By comining 9,, 3 an 4 it follows that [V ut, ] = U T QxU x, 5 where for all x Qx := sxfx Q x Q x sxax sxx 6 sxcx with Q x : = sxx [sxax] [sxx], x x Q x : = sxex [sxx] [sxcx]. x x A. Spacing functions If Qx for all x, then the time erivative in 5 is clearly non-positive for all t >. However, such a conition on Q is conservative. To ecrease that conservatism, we introuce matrix value functions Υ i such that U T Υ i xu x = an, therefore, can e ae to Q without altering the integral. Υ i are calle spacing functions. We parameterize Υ i y polynomials p i. The iea came from [7]. The following hols for any p R[x], ecause of the ounary conitions 6. [up xu] x x = up xu x = x = x =. 7 Using the chain rule, we have x [up xu] = u x [p x]u + p xuu x = U T Υ xu, 8 where for all x x [p x] p x Υ x :=. 9 Comining 7 an 8 results in U T Υ xu x =. 3 Likewise in 7, ecause of the ounary conitions 6, the following hols for any p R[x]. [up xu] x =. x Following steps of 8 for x [up xu], gives x [p x] p x Υ x := 3 such that U T Υ xu x =. 3 Similarly to 7, the following is true for any p 3 R[x]. [up 3 xu x ] x =. 33 x Note that the left-han sie of 33 can e written as follows. [up 3 xu x ] x x = u x p 3 xu x + u [p 3 x]u x x x + up 3 xu x x x, 34 where we nee property. Applying integration y parts to the secon integral of the right-han sie of the last equation in 34 results in up 3 x [u x ] x x = up 3 xu x x = x = u x [up 3 x] x x x = u x u x p 3 x + u [p 3 x] x. x 35

5 From 34 an 35 the following hols. [up 3 xu x ] x x = u [p 3 x]u x x u [p 3 x]u x x x = U T Υ 3 xu x, 36 where for all x x [p 3 x] x [p 3 x] Υ 3 x :=. 37 Comining 33 an 36 gives U T Υ 3 xu x =. 38 Following steps for x [up 4 xu x ] with any p 4 R[x], leas to the following. U T Υ 4 xu x =, 39 where for all x x [p 4 x] x [p 4 x] Υ 4 x :=. 4 From 5, 3, 3, 38 an 39 the following hols. 4 [V ut, ]= U T Qx+ Υ i x Ux. 4 i= By sustituting for Q an Υ i from 6, 9, 3, 37 an 4 in 4 we can efine M = Φa,, c,, e, f, s, p, p, p 3, p 4, 4 if for all x M x M x M 3 x Mx = sxax sxx, 43 sxcx where M x := sxfx + [p x] + [p x], x x M x := sxx [sxax] [sxx] x x + p x + [p 3 x p 4 x], x M 3 x := sxex [sxx] [sxcx] x x + p x + [p 4 x p 3 x], 44 x such that [V ut, ] = U T MxUx. 45 Theorem : Suppose that for 5 there exist s, p i R[x], for i =,..., 4, an θ >, such that sx θ an Mx for all x, where M is efine as in Then 5 is stale. Proof: If conitions of Theorem are satisfie, let a = inf {sx}, = sup{sx}. 46 x x Since sx θ > for all x, then, a > an the following hols for all v L. a v L = inf v x x sxv x x x sup{sx} x v x x = v L. 47 Using 7 we have a v L V v v L. Since Mx, from 45 it follows that [V ut, ] for all t >. Theorem with a,, efine as in 46, m = l = an c = ensures staility of 5. Theorem 3: Suppose that for 5 there exist θ, γ > an s, p i R[x], for i =,..., 4, such that sx θ an Mx + γsx for all x, where M is efine as in 4 44 an Sx := sx. 48 Then for all t > solution to 5 satisfies ut, L a u, L exp{ γ t}, 49 where a, are efine as in 46. Proof: Uner the assumptions of Theorem 3, 47 hols. With 48 an we can write V ut, = U T SxU x. 5 Since Mx + γsx for all x, it hols that U T Mx + γsxu x. 5 Since γ is a scalar, 5 can e easily satisfie as follows. U T MxUx γ U T SxUx, which with 45 an 5 provies [V ut, ] γv ut,. Using proof of Theorem with c/ = γ, results in 49. V. LYAPUNOV STABILITY IN TERMS OF SOS OPTIMIZATION Solving optimization over polynomials is computationally har. Thus, in this section we present SOS programming alternatives, whose solutions yiel solutions to prolems in Theorems an 3 an, therefore, provie sufficient conitions for staility of System 5.

6 Theorem 4: Suppose that for 5 there exist s, p i R[x] for i =,..., 4, θ >, n, n Σ[x] an Q, Q, Q 3 Σ[S 3 x] such that for all x, x, sx = θ + x x n x + x x n x, Mx = Q x x x Q x x x Q 3 x, 5 where M is efine as in Then 5 is stale. Proof: If 5 hols, then clearly sx θ an Mx for all x. Using Theorem provies staility of 5. Theorem 5: Suppose that for 5 there exist θ, γ >, s, p i R[x] for i =,..., 4, n, n Σ[x] an Q 4, Q 5, Q 6 Σ[S 3 x] such that for all x, x, sx = θ + x x n x + x x n x, Mx + γsx = Q 4 x x x Q 5 x x x Q 6 x, 53 where M is efine as in 4 44 an S as in 48, then for all t > solution to 5 satisfies ut, L a u, L exp{ γ t}, 54 where a, are efine as in 46. Proof: If 53 is true, then for all x, sx θ an Mx + γsx, which, comine with Theorem 3, gives 54. VI. NUMERICAL VERIFICATION OF THE PROPOSED METHOD A. KISS moel We applie the metho escrie in Sections IV an V to staility of the iological KISS PDE name after Kierstea, Slookin an Skelam, which escries population growth on a finite area. For more etails see [3]. The system is moele y the following PDE. u t = h u x x + u x x + ru, 55 where h, r >, x R an scalar function u satisfies zero Dirichlet ounary conitions. It is claime in [3] that if is a square with ege of length l, then l cr := π h r 56 efines a critical length. That means, if l > l cr, then 55 is unstale. Alternatively, for given l an r 56 efines h cr as h cr := l r/π. 57 Therefore, if h < h cr, then 55 is unstale. For our purpose we fix l = an aritrarily choose r = 4. Thus, accoring to 57, h cr.3. Using a isection search over h, we etermine minimum h cr for which SOS prolem in Theorem 4, with ax = h, x =, cx = h, x =, ex =, fx = 4 for all x 58 log ut, L egs=4 egs=6 egs=8 egs= egs= Numerical t Fig.. Semi-log plots of the L norm of the numerical solution to 55 with u, x = 3 x x x x an ouns, given y the propose metho with ifferent egs. TABLE I MINIMUM h cr VS DEGs FOR 55 egs analytic h cr may e shown to e feasile. Results for ifferent egrees of s egs are presente in Tale I. Now we choose l =, h = an r = 4. Using a isection search over γ, we etermine the maximum γ for which the SOS prolem in Theorem 5, with 58, may e shown to e feasile. Results for ifferent egs are presente in Tale II. Using finite ifference scheme, we numerically solve 55 with u, x = 3 x x x x. Plots of log ut, L versus t, using a numerical solution, an ouns on log ut, L, given y the propose metho for ifferent egs, are presente in Fig.. These plots allow us to etermine γ y examining the rate of ecrease in the L norm. Plots are aligne at t = in orer to etter compare our SOS estimates of γ to the estimate of γ erive from numerical simulation as a function of increasing egs. B. Ranomly generate system Consier u t = 5x 5x x + 3x u xx + u x x + x 5x u x + 5x + 6x u x 7x 4 3x 5x 8x 3 5x 4 u, u, x = 3 x x x x 59 where x :=, an the scalar function u satisfies zero Dirichlet ounary conitions. Using a isection search over γ, we etermine maximum γ for which SOS prolem in Theorem 5, with ax = 5x 5x x + 3x, ex = 5x + 6x, cx = 5x 5x x + 3x, x = x 5x, fx = 7x 4 3x 5x 8x 3 5x 4, x = may e shown to e feasile. TABLE II MAXIMUM γ VS DEGs FOR 55 WITH h = egs γ

7 log ut, L Propose metho Numerical simulation t Fig.. Semi-log plots of the L norm of the numerical solution to 59 an oun, given y the propose metho with egs = 8. Using finite ifference scheme, we numerically solve 59. The estimate rate of ecay, ase on numerical solution, is 3.7. The compute rate of ecay, ase on our SOS metho, is.5 for egs = 8. Plots are given in Fig. an, as for Fig., are aligne at t =. VII. CONCLUSION AND FUTURE WORK In this paper we have presente a metho which allows us to search for Lyapunov functionals for paraolic linear PDEs with two spatial variales an spatially epenent coefficients. We emonstrate accuracy of the metho in estimating critical iffusion for the KISS PDE an the rate of ecay for the KISS PDE an for a ranomly chosen PDE. In future work, we will exten the metho y consiering Lyapunov functionals of more complicate types, e.g. functionals with semi-separale kernels, as in [3]. 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