Introduction

Here we are interested in the following equation:

Above, a, b, c are continuous functions, where, for c, there exist C_c, m > 0 such that

The increasing set 0 < t₁ < t₂ < · · · < t_k < · · · satisfies t_k → ∞ when k → ∞; ck are real numbers, = 1, 2, . . . , such that there exists q > 0 for which

Here n > 1. We assume that the initial data(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(0, x) of the amount of the gas in this instant may be less precise than the measurement (0, 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 setcover many well-known physical phenomena such as: problems with periodic conditions u(0) = u(t₁), problems with Bitsadze–Samarskii conditions, 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 given in a feedback form, i.e.,(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 p The problem of exponential stability associ- ated to (1), i.e., whether the solution of the uncontrolled equation (1) ( 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 blow- up 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 L² norm, but also in the H¹-norm.

Let then we have

In (1), let us perform the transformation

Recall that n > 1. We equivalently express (1) in terms of . as

Her

And

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

For the particular case p = 2, we denote by and bythe standard norm and scalar product in L ² (0, 1), respectively. While for 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 Hr (0, 1), , stand for the standard Sobolev spaces on (0, 1), while H1 0 (0, 1) restricts to null trace functions. We recall that, via the Poincaré inequality, we have that the norm kd/dx · k is an equivalent norm in H (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 α₁ = β₁ = 1 and α₂ = β₂ = 0, we have that A is self-adjoint and has a countable set of simple real eigenvalues with the corresponding eigenfunctions. Moreover, the eigenvalues can be arranged as an increasing sequence with λ_k → ∞, and the eigenfunctions set forms an orthonormal basis in L² (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 , we have that the eigenvalues of A satisfy lim_j→∞ λ_j/j² = const. Hence, we have

The energy space H_Aof a positive definite self-adjoint operatorL²(0,1)→L²(0,1)is a Hilbert space defined by introducing the inner product〈f,g〉_A:=〈Af,g〉and the energy norm The resulting space is then completedby including all limit elements. Moreover, the eigenfunctions system ofAform a com-plete system in L² (0,1)andHA. For details, see, e.g., [3]. It is clear that for ζ >0 sufficiently large, the operator (Ibeing the identity operator) is self-adjoint andpositive definite. Thus, operator A=A+ ζI introduces an energy space,HA, whichis in fact the space(0,1). The eigenvalues set of A+ζI is, while the corresponding eigenfunctions are the same as ofA, namely Moreover, invirtue of the results in [3], the set{φk}k∈N∗is complete inH10(0,1), and there exists 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 . into the equations via the Dirichlet map defined as (see [7, Eq. (2.16)]): for large enough, we denote bythe solution to the equation

Notice that γ > 0 sufficiently large guarantees the unique existence of such solution z. The dual of D_γ, see [7, Eq. (2.17)] and [7, Ex. 2.4], depends on, which, in our case, is given by= 1, 2, . . .

Let

N are the constants sufficiently large such that for each of them, the corresponding equa- tion (11) has a unique solution D_γk , k = 1, 2, . . . , N . Then, following the notations in [7, Eqs. (2.20)–(2.26)], we denote by B the Gram matrix

and multiply it on both sides by

to define

Then we introduce the matrix

and set the following feedback forms:

Finally, we introduce u. as

Here_N stands for the Euclidean scalar product in 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).

Lemma 1. The solution z of

can be written in a mild formulation as

where the kernel

for(0,1) Here

and

The quantities q_ji(t) and(t), involved in the definition of p, obey the estimates. for some C. > 0 depending on N,

for all i ,j= 1, 2, . . . , N and j = N + 1, N + 2

Relying on the key lemma abiove , 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 by the following integral operator

We claim that the operator I - 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

together with relations (17)–(18), it yields from above that

where C₁>0 is some constant. We bound the RHS of the above relation by taking intoaccount that

becausewhenandNis large enough; and by taking into account that,due to (7), we have is a convergent series. It yields that

Clearly seen, for γ₁, N large enough, the above quantity can be made arbitrary small

We go on following [5] and introduce the quantities

where

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 γ₁, N sufficiently large, we may assume that M_∞(V₊) are small enough such that

Thus, relation [5, Eq. (1.7)] (or, equivalently, [5, Eq. (1.5)]) is satisfied. Consequently, we are in power to apply the result in [5, Thm. 1.1], and we deduce that the operator I- is invertible with bounded inverse.

Now, returning to (20), we express the nonlocal initial condition as

Thus, existence of a solution . is equivalent with the fact that the map G, defined as

has a fixed point. Here

In the next section, we aim to prove the main result of the present work stated below.

Theorem 1. Let r > 0 sufficiently small. Then there exists a unique fixed point y ∈ B_r(0) of the map G : Y → Y, where B_r(0) is the ball centered at the origin of radius r of the space

In particular, once we plug the feedback controller

into equation

it yields that its unique solution is exponentially decaying in the H.-norm. Here are given im (12)-(14). while ...,N, are the firs N eigenfuncions of the operator A.

Note that, due to the linearity of the control . and the definition of the transformation , we have

Proof of the Theorem 1

It is clear that for all . we have

We need to estimate thenorm of Gy. So, in particular, we need to estimate the |·|Y - norm of for. We begin with the L² -norm of. 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 as

Where

We will use the well-known Sobolev embedding

and the fact that is an equivalent norm in (0, 1)

Below, will stand for different constants that may change from line to line, however, we keep denote them by 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

where using the Sobolev embedding, we deduce that

It follows by (22) and the fact that −.. + nN + 1.4 < 0 for .. large enough that

for all

Again involving Parseval’s identity, we get

In virtue of Schwarz’s inequality, we obtain

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 eigenfunc- tions. Recalling relation (7), we see that the above series converge. Thus,

where we used relation (22) and the fact that 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

Consequently, we have

We know by [5] that (I - )⁻¹ is bounded, so, it follows from above together with (30) that

We go on with the estimates in the H¹-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 thatlarge enough, we see that

Next,

Then, using the obvious inequality

we arrive at

Arguing as in (24)-(27), we get

Finally, recalling relation (9)

Then, in virtue of (19) and arguing like before, we have

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

Now, gathering together relations (30), (31), (40), (41) and observing that the beta functions .(1/4, 3/4), (1/4, 1/4) are finite, we obtain that

for some positive constant ..

Similar computations lead to

Next, we set In virtue of (42) and (43), we get that for for r sufficiently small. Hence, maps the ball B_r(0) into itself. Then for, we have with for r sufficiently small. Thus, 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

Returning to the transformation the conclusion of the theorem follows imme- diately.

Conclusions

Here we discussed about the semilinear heat equation on the rod with polynomial non- linearity 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, 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 wn, one can consider the term 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].

Acknowledgments

This research was supported by a grant of the Romanian Ministry of Research and Innovation, CNCS–UEFISCDI, project number PN-III-P1-1.1-TE-2019-0348 within PNCDI III

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