Feedback exponential stabilization of the semilinear heat equation with nonlocal initial conditions

Abstract. The present paper is devoted to the problem of stabilization of the one-dimensional semilinear heat equation with nonlocal initial conditions. The control is with boundary actuation. It is linear, of finite-dimensional structure, given in an explicit form. It allows to write the corresponding solution of the closed-loop equation in a mild formulation via a kernel, then to apply a fixed point argument in a convenient space.

Here n > 1. We assume that the initial data w(0, ·) is square integrable on (0, 1). Finally, u w is a boundary actuator. Parabolic problems with nonlocal initial conditions, as (1), appear in the modelling of concrete problems such as heat conduction and in thermoelasticity. For example, if there is too little gas at the initial time, then the measurement w(0, x) of the amount of the gas in this instant may be less precise than the measurement w(0, x) + ∞ k=1 c k w(t k , x) of the sum of the amounts of this gas (for details, see [4]). They can be as well used for modelling certain physical measurements performed repeatedly by the devices having relaxation time comparable to the delay between the measurements. Particular cases of the set {c k } ∞ k=1 cover many well-known physical phenomena such as: problems with periodic conditions u(0) = u(t 1 ), problems with Bitsadze-Samarskii conditions u(0)+c 1 u(t 1 ) = c 2 u(t 2 ), regularized backward problems, etc. (for more details, see [11] and the references therein).
In this work, we address the problem of asymptotic exponential stabilization of (1). More precisely, we look for a control u w given in a feedback form, i.e., u w (t) = u w (w(t)), such that once inserted into equation (1) it yields that the corresponding solution of the closed-loop equation (1) satisfies the exponential decay for a constant C > 0 and arbitrarily large ρ. The problem of exponential stability associated to (1), i.e., whether the solution of the uncontrolled equation (1) (u w ≡ 0) satisfies an exponential decay as above, has been addressed in many works, see, for example, [2,4,9]. There it is shown that, under some appropriate conditions on the coefficients, the solution decays exponentially fast at infinity. In our case, since we let free the coefficients a, b, c in (1), it is clear that one cannot expect such an exponential decay to hold true. In fact, we do not even know whether equation (1) has solutions at all. But if it has, the blowup phenomenon may occur (see [6]). Equation (1) is in connection with the stabilization to states for the semilinear heat equation with nonlocal initial conditions. We stress that the time dependency of the linear governing operator is related to nonstationary states. In other words, we include in our study the case of stabilization to trajectories for the semilinear heat equation. The ideas in this paper rely on the controller design technique developed in [7]. There is an explicit feedback form control designed for stabilizing parabolic-type equations. Its simple form allows us to write the corresponding solution of the closed-loop equation in a mild formulation via a kernel similar to the heat kernel. In this way the solution becomes a fixed point of a nonlinear map. Then, applying a fixed point argument, in a convenient space, we prove simultaneously the well-posedness of the equation and the exponential decay of the solution not only in the L 2 -norm, but also in the H 1 -norm.
Let δ > max x∈[0,1] |b(x)|, then we have In (1), let us perform the transformation Recall that n > 1. We equivalently express (1) in terms of y as Let us notice that, in virtue of relations (4) and (2), we have by using the obvious inequality (which will be frequently used below as well) Besides this, by (3) it is clearly seen that the series We set L p (0, 1) = f : (0, 1) → R: For the particular case p = 2, we denote by · and by ·, · the standard norm and scalar product in L 2 (0, 1), respectively. While for p = 2, we denote which is a norm in L p . Below, we denote by y the derivative with respect to x, i.e., y (x) = (d/dx)y(x).

Let us denote
Here H r (0, 1), r ∈ N * , stand for the standard Sobolev spaces on (0, 1), while H 1 0 (0, 1) restricts to null trace functions. We recall that, via the Poincaré inequality, we have that the norm d/dx · is an equivalent norm in H 1 0 (0, 1). In this work, our results rely on the special properties of the spectrum of the operator A. This is related to the Sturm-Liouville theory. In virtue of the results in [1, Sect. 2.4.1], considered for the particular case: definition interval (0, 1), function p = 1, r(x) = a(x)+α, constants α 1 = β 1 = 1 and α 2 = β 2 = 0, we have that A is self-adjoint and has a countable set of simple real eigenvalues {λ k } k∈N * with the corresponding eigenfunctions {ϕ k } k∈N * . Moreover, the eigenvalues can be arranged as an increasing sequence with λ k → ∞, and the eigenfunctions set forms an orthonormal basis in L 2 (0, 1). Since λ k → ∞ when k → ∞, we see that for N ∈ N large enough, λ k > 0 for all k N . Besides this, by [12, Thm. 4.3.1 (7)] considered for the particular case: definition interval (0, 1), p = 1, q(x) = −a(x) − α, w = 1, A = 1 0 0 0 , and B = 0 0 1 0 , we have that the eigenvalues of A satisfy lim j→∞ λ j /j 2 = const. Hence, we have The energy space H A of a positive definite self-adjoint operator A : D(A) ⊂ L 2 (0, 1) → L 2 (0, 1) is a Hilbert space defined by introducing the inner product f, g A := Af, g and the energy norm f A := f, f A . The resulting space is then completed by including all limit elements. Moreover, the eigenfunctions system of A form a complete system in L 2 (0, 1) and H A . For details, see, e.g., [3]. It is clear that for ζ > 0 sufficiently large, the operator A + ζI (I being the identity operator) is self-adjoint and positive definite. Thus, operator A = A + ζI introduces an energy space, H A , which is in fact the space H 1 0 (0, 1). The eigenvalues set of A + ζI is {λ k + ζ} ∞ k=1 , while the corresponding eigenfunctions are the same as of A, namely {ϕ k } k∈N * . Moreover, in virtue of the results in [3], the set {ϕ k } k∈N * is complete in H 1 0 (0, 1), and there exists C > 0 such that Recall that λ j → ∞ when j → ∞. So, we assume that N is large enough such that It is easy to see that the fundamental solution associated to −A, namely can be written as and, by [10], we know that for some positive constant C.

Construction of the stabilizer and the main result
We will apply the control design technique described in [7,Chap. 2]. The first idea is to lift the boundary control u into the equations via the Dirichlet map defined as (see [7, Eq. (2.16)]): for β ∈ R and γ > 0 large enough, we denote by D γ β := z the solution to the equation Notice that γ > 0 sufficiently large guarantees the unique existence of such solution z.
N are the constants sufficiently large such that for each of them, the corresponding equation (11) and multiply it on both sides by Then we introduce the matrix and set the following feedback forms: Here ·, · N stands for the Euclidean scalar product in R N . For more details on the construction of u, one can see [7,Ex. 2.5].
Next, we plug this feedback into equation (5) and argue similarly as in [7, Eqs. (2.27)-(2.29)] in order to equivalently rewrite (5) as an internal-type control problem as follows: Following the ideas in [7, Lemma 7.1], we may arrive to the following result related to the linear operator that governs equation (15). can be written in a mild formulation as where the kernel for t 0, x, ξ ∈ (0, 1). Here and The quantities q ji (t) and w j i (t), involved in the definition of p, obey the estimates: for some C q > 0 depending on N , for all i, j = 1, 2, . . . , N ; and for some C w > 0 depending on N , for all i = 1, 2, . . . , N and j = N + 1, N + 2, . . . . Moreover, for all z 0 ∈ L 2 (0, 1), we have that Relying on the key lemma above, we rewrite (5) in a mild formulation as where p is defined in (16). We aim to express, in (20), the nonlocal initial condition inherited from equation (5). To this end, we denote byK the following integral operator https://www.journals.vu.lt/nonlinear-analysis K : L 2 (0, 1) → L 2 (0, 1): We claim that the operator I −K is invertible. To show this, we apply the result in [5, Thm. 1.1]. The main ingredient we will use is the exponential decay of the kernel p. In the spirit of [5], let us denote Then we show that Indeed, taking into account the particular form of K, we have Making use of the uniform bound of the eigenfunction system, the fact that K(x, ξ) dξ where C 1 > 0 is some constant. We bound the RHS of the above relation by taking into account that because λ i → ∞ when i → ∞ and N is large enough; and by taking into account that, due to (7), we have ∞ j=N +1 1/|λ j − γ 1 | is a convergent series. It yields that Clearly seen, for γ 1 , N large enough, the above quantity can be made arbitrary small. We go on following [5] and introduce the quantities It is easy to see that with similar arguments we used to obtain (21), based on the uniform bounds of the eigenfunction system, by taking γ 1 , N sufficiently large, we may assume that M ∞ (V ± ) are small enough such that In the next section, we aim to prove the main result of the present work stated below.
In particular, once we plug the feedback controller it yields that its unique solution is exponentially decaying in the H 1 -norm. Here A, Λ γ k are given in (12)-(14), while ϕ k , k = 1, 2, . . . , N , are the first N eigenfunctions of the operator A.
Note that, due to the linearity of the control u and the definition of the transformation w → y, we have y(t, 0) = u y (t) is equivalent with w(t, 0) = u w (t).

Proof of the Theorem 1
It is clear that for all y ∈ Y, we have We need to estimate the |·| Y -norm of Gy. So, in particular, we need to estimate the |·| Ynorm of Fy for y ∈ Y. We begin with the L 2 -norm of Fy. We aim to use the Parseval's identity. In order to do this, based on the kernel's form (16), we conveniently rewrite the term Fy as We will use the well-known Sobolev embedding H 1 (0, 1) ⊂ L p (0, 1) ∀p > 1 and the fact that d/dx · is an equivalent norm in H 1 0 (0, 1). Below, C will stand for different constants that may change from line to line, however, we keep denote them by C for the ease of notations. It follows via the Parseval's identity and the fact that the eigenfunctions are uniformly bounded and relation (6) that Involving relation (17) and the Schwarz's inequality, it yields It follows by (22) and the fact that −γ 1 + nN + 1/4 < 0 for γ 1 large enough that for all t s 0. Again involving Parseval's identity, we get In virtue of Schwarz's inequality, we obtain 2 (s, ξ)e 2(n−1)N s y 2(n−1) (s, ξ) dξ Then, making use of (10), we get Taking advantage of (6) and the Sobolev embedding, we arrive at in virtue of relation (22). Next, by taking into account relations (6), (18) and the uniform boundedness of the eigenfunctions. Recalling relation (7), we see that the above series converge. Thus, where we used relation (22) and the fact that −γ 1 + nN + 1/4 < 0 for γ 1 large enough. Hence, it follows by (23)-(29) that where B(x, y) is the classical beta function. By the exponential semigroup property we have that 1 0 p(t, ·, ξ)y 0 (ξ) dξ Ce −γ1t y 0 ∀y 0 ∈ L 2 (0, 1).

Consequently, we have
We know by [5] that (I −K) −1 is bounded, so, it follows from above together with (30) that We go on with the estimates in the H 1 -norm. We have, in virtue of (8), that Involving relation (17), the Schwarz's inequality, and the Sobolev embedding, we obtain Then by (22) and the fact that −γ 1 + nN + 3/4 < 0, for γ 1 large enough, we see that Next, (s, ξ)y n (s, ξ)ϕ j (ξ) dξ (s, ξ)y n (s, ξ)ϕ j (ξ) dξ Then, using the obvious inequality we arrive at Arguing as in (24)-(27), we get Finally, recalling relation (9), (s, ξ)y n (s, ξ)ϕ i (ξ) dξ Then, in virtue of (19) and arguing like before, we have Ce −γ1(t−s) sup l=1,2,...,N c(s, ·)y n (s, ·), ϕ l (·) Therefore, (32)-(39) imply that Heading towards the end of the proof, we note that since the presence of the λ j in the infinite summation is controlled as in (35) by the presence of t 1/2 . Consequently, via the semigroup property, we deduce that So, arguing as in (31) and using (40), we arrive at Next, we set B r (0) := {y ∈ Y: |y| Y r}, r > 0. In virtue of (42) and (43), we get that for y ∈ B r (0), |Gy| Y Cr n r for r sufficiently small. Hence, G maps the ball B r (0) into itself. Then for y 1 , y 2 ∈ B r (0), we have |Gy 1 − Gy 2 | Y nCr n−1 |y 1 − y 2 | Y with nCr n−1 < 1 for r sufficiently small. Thus, G is a contraction on B r (0). We conclude by the contraction mapping theorem that G has a unique fixed point in B r (0) when r is sufficiently small. This implies that equation (15) has a unique solution, which satisfies |y| Y r < ∞.
Returning to the transformation y → w, the conclusion of the theorem follows immediately.

Conclusions
Here we discussed about the semilinear heat equation on the rod with polynomial nonlinearity and with nonlocal initial conditions. We addressed the problem of boundary exponential stabilization, but in the same time, we showed the well-posedness of the model since there was no result guaranteeing the existence of solutions. The exponential decay is of order e −N t , where N is some natural number standing for the dimension of the controller. It is clear that taking N large enough, the exponential decay can be made arbitrarily fast, but with large dimension of the controller. We stress that, instead of the nonlinearity w n , one can consider the term w∂ x w, and with slight adjustments the proof is similar. It remains as an open problem the multidimensional space case. In this case the problem comes from the estimates of the eigenfunction system and for the fundamental solution (10), which are not as good as in the one-dimensional case. For the two-dimensional case, one can argue for cubic nonlinearities as in [8].