Solvability, Completeness and Computational Analysis of A Perturbed Control Problem with Delays

Abstract As a first step, we provide a precise mathematical framework for the class of control problems with delays (which we refer to as the control problem) under investigation in a Banach space setting, followed by careful definitions of the key properties to be analyzed such as solvability and complete controllability. Then, we recast the control problem in a reduced form that is especially amenable to the innovative analytical approach that we employ. We then study in depth the solvability and completeness of the (reduced) nonlinearly perturbed linear control problem with delay parameters. The main tool in our approach is the use of a Borsuk–Ulam type fixed point theorem to analyze the topological structure of a suitably reduced control problem solution, with a focus on estimating the dimension of the corresponding solution set, and proving its completeness. Next, we investigate its analytical solvability under some special, mildly restrictive, conditions imposed on the linear control and nonlinear functional perturbation. Then, we describe a novel computational projection-based discretization scheme of our own devising for obtaining accurate approximate solutions of the control problem along with useful error estimates. The scheme effectively reduces the infinite-dimensional problem to a sequence of solvable finite-dimensional matrix valued tasks. Finally, we include an application of the scheme to a special degenerate case of the problem wherein the Banach–Steinhaus theorem is brought to bear in the estimation process.


Introduction
Linear control problems with nonlinear functional perturbations are of great interest in modern studies owing to their nontrivial mathematical structure and wide applications in diverse fields. Of special interest are nonlinearly perturbed control problems [11,25,26] with delay parameters, modeling some real situations in widely used remote control systems. The solvability and reliability of such control problems strongly depends on the topological structure [1,3,4,8,10,11,17,15,23] of the corresponding solution set, its completeness and stability. Here we study in detail the solvability, completeness and topological structure of the corresponding solution set for a suitably reduced linear control problem with nonlinear functional perturbation, depending on delay parameters, using generalized fixed point [9,18,19,20,21,22] theory, which enabled us to advance the body of knowledge found in the literature. We investigate its classical analytical solvability under some special nonlocal conditions [2,7], imposed on the linear control and nonlinear functional perturbation, and study the feasibility of a naturally related control problem computational scheme, based on the classical projection discretization method, which reduces the infinite dimensional problem to a sequence of solvable finite dimensional matrix valued tasks. The solvability and completeness of the (reduced) nonlinearly perturbed linear control problem with delay parameters is studied in detail making use of a Borsuk-Ulam type fixed point theorem that is particularly well-suited for analyzing the general topological structure of certain kinds of reduced control problem solutions, with a focus on estimating the dimension of the corresponding solution set, and proving its completeness. We also investigate the related classical analytical solvability under some special, mildly restrictive, conditions imposed on the linear control and nonlinear functional perturbation. As its application oriented part there is described a novel computational projection-based discretization scheme for obtaining accurate approximate solutions of the control problem along with useful error estimates. The numerical scheme effectively reduces infinite-dimensional problem to a sequence of solvable finite dimensional matrix valued tasks. In addition, we include an application of the scheme to a special degenerate case of the problem wherein the Banach-Steinhaus theorem is brought to bear in the estimation process.
Let U ⊂ L ∞ ([0, 1]; Y ) be a suitable control subspace. Then we say that the dynamical system (2.1) -(2.2) is completely controllable on the subspace U ⊂ L ∞ ([0, 1]; Y ), if it has a solution x ∈ AC 1 ([0, 1]; X) and u ∈ U for all x 0 , x 0 and x 1 ∈ X. Next, we study this controllability problem as that of describing the existence of the corresponding solution set of (2.1) -(2.2) and its topological structure.
The problem (2.1) -(2.2) can be recast as the reduced control system for which for all t ∈ [0, 1]. Now, based on the representation (2.1a) -(2.1b), one can define a linear operator A : allowing to rewrite the control problem (2.1a) -(2.1b) as the following equivalent operator equation: Consequently, we need to study the structure of the solution set N (A, where the linear operator A : E 1 → E 2 is assumed to be a closed, surjective mapping from the Banach space where the linear operator B : L ∞ ([0, 1]; Y ) → X acts as for any u ∈ L ∞ ([0, 1]; Y ). Having assumed that the condition (2.7) holds, we easily obtain the result. Consider now the mapping (2.5) and assume additionally that its domain domF =domA ∩ S r (0), where S r (0) ⊂ E 1 is a sphere of radius r > 0, centered at 0 ∈ E 1 . We need the following [9,18,11] useful definitions. Definition 2.2. A mapping F : E 1 → E 2 from a Banach space E 1 to a Banach space E 2 is called A-compact subject to a linear operator A : E 1 → E 2 , if it is continuous and for any bounded sets A 1 ⊂ domF and A 2 ⊂ E 2 the set F(A 1 ∩ A −1 (A 2 )) ⊂ E 2 is relatively compact in E 2 (the empty set ∅ is considered, by definition, compact).
Given a continuous nonlinear mapping F : E 1 → E 2 , domF ⊂ S r (0), and a closed, surjective linear operator A : E 1 → E 2 , one can also define the following numerical characteristics: where k(A) := || A −1 || and the operator A := A| E1/ ker A is an invertible continuous linear operator from the factor-space Introduce preliminarily the following definition. Then, using a generalized Borsuk-Ulam type fixed point theorem [9,18], one can formulate the following result that characterizes the solution set N (A, F) ⊂ AC 1 ([0, 1]; X) × L ∞ ([0, 1]; Y ) of the reduced control problem (2.4).
Theorem 2.4. Let the linear closed operator A : E 1 → E 2 , defined by the expression (2.3), satisfy the dimension condition dim ker A ≥ 1, the continuous mapping F : E 1 → E 2 , defined by the nonlinear expression (2.5), be A-compact and the inequality k(A) < k F hold. Then the reduced problem (2.4) is solvable in the domF ⊂ S r (0), with a nonempty solution set Using the above result, one can easily check that for the reduced control problem (2.4) and for any sphere S r (0) ⊂ E 1 , r > 0, Taking the statement above into account, we will study the topological structure of the solution set N (A, are endowed with some additional analytical structure. Namely, we consider the case, when for the closed linear operator satisfies the complete controllability condition (2.6).

Classical solution set analysis
Consider a slightly generalized nonlocal evolution control problem (2.1) in the following form: where boundary conditions are chosen in following functional form: and We prove two theorems on the controllability of problem (3.1) -(3.4). In particular, we will show that the corresponding solution set to the problem (3.1) -(3.4) is nonempty and has a nontrivial topological dimension on the control space U , thus proving the problem controllability. For this purpose we apply the adapted controllability of semilinear control systems of the first order with constant time-delay equation control in [16] and the classical Banach contraction mapping theorem.
In Section 3, we use the following assumption: Assumption (A). Operator A is the infinitesimal generator of a strongly continuous cosine family {C(t) : t ∈ R} of bounded linear operators from X into itself.
Recall that the infinitesimal generator of a strongly continuous cosine family C(t) is the operator A : X ⊃ D(A) → X defined by The associated sine family {S(t) : t ∈ R} is defined by the integral expression Following [6,16], we can derive that the mild solution x(·) to the system (3.1) -(3.4) satisfying the equation Moreover, it is clear that the function x(·) belongs to the class C 1 ([0, T ], X). We shall apply the notation where * denotes the adjoint. Moreover, let . Now, we define a real Banach space by Moreover, we define an operator F : X −→ X by where F 1 : X −→ B 1 is given as

192
Solvability, Completeness and Computational Analysis of A Perturbed Control Problem with Delays x (a 2 (s))) ds.
To prove the controllability of system (3.1) -(3.4) we need the following result.
if and only if for every initial state x 0 ∈ E and a final state x T ∈ X, the operator F : X −→ X given by (3.6) -(3.9) has a fixed point, i.e. there is some (x, u) ∈ X such that F(x, u) = (x, u).
Proof. Let the system (3.1) -(3.4) be controllable. Then there exists a control function u(·) ∈ B 2 , which steers the state of the system given in equation (3.5) from x 0 to x T . That is s)), x (s), x (a 2 (s))) ds.
From the above equation and from (3.9), we obtain We then choose a function u(·) satisfying (3.10) as Now, if we compare (3.11) with (3.8), we see that F 2 (x, u) = u. Moreover, with this control function, the corresponding solution given in (3.5) reduces to equation (3.7). Consequently, F 1 (x, u) = x. Therefore, F(x, u) = (x, u), i.e. F has a fixed point. Conversely, assume now that the operator F has a fixed point, i.e., F(x, u) = (x, u) for some (x, u) ∈ X. We want to show that there exists some control function u(·) ∈ B 2 such that x(T ) = x T . Since F(x, u) = (x, u) then, by (3.7) and (3.8), we obtain the formulas x (a 2 (s))) ds In order to get x(T ) = x T , we put t = T in (3.12) and apply (3.9). Consequently, we have Moreover, problem (3.1) -(3.4) is controllable on [0, T ], so the proof of Theorem 3.2 is complete.
As a consequence, we show the nontriviality of the topological dimension of the corresponding solution set N (A, F).
Consequently, F : X −→ X is a contraction mapping, so Banach's theorem implies that F has a unique fixed point in X. Hence, by Theorem 3.2, the system (3.1) -(3.4) is controllable on [0, T ], which completes the proof.
In particular, if x 0 = 0, p = 1, t 1 = T, c := −c 1 ε := ε 1 , then Theorems 3.2 and 3.3 are reduced to the theorems, where the nonlocal condition (3.2) is of the form respectively. A similar remark is also true for the nonlocal condition (3.3).

Computational Scheme and its Stability
We are now interested in describing a feasible numerical computational scheme, based on the classical projection method and suitable for solving the nonlinear control problem, studied above. Consider the operator equation where w ∈ X is its solution. Let nowX N ⊂X N +1 ⊂ X andỸ N ⊂Ỹ N +1 ⊂ Y, N ∈ Z + , are suitable approximating finitedimensional Banach subspaces, P (x) N : X →X N , N ∈ Z + , and P (y) N : Y →Ỹ N , N ∈ Z + , are the corresponding projectors. Now we consider a countable sequence of equations P (y) on elementsw N ∈X N , N ∈ Z + , which are suitable approximations to a searched solution of equation (4.1), being in general non-unique, as dim kerA ≥ 1. Note here that the projection method is called "realizable", if the set M ⊂ X of solutions to equation (4.1) is nonempty, and for enough large N ∈ Z + there are nonempty sets M N ⊂X N of solutions to equations (4.2). The method is called "convergent" if it is realizable and there is fulfilled the condition It is obvious that for practical applications the realizability criteria of the projection method and its convergence are very important, therefore we will analyze them making use of modified version of Theorem 2.4. Namely, we assume that the necessary condition dim kerA ≥ 1 subject to the parameters of the whole control space U is satisfied. Then the following result characterizes the realizability of the related computational scheme subject to the discrete approximations (4.2).
Then for sufficiently large integers N ≥ N 0 ∈ Z + the solution setsM N ⊂X N are nonempty and the convergence condition (4.3) holds.
and choose such integer N 0 ∈ Z + , that dim ker(P (y) N0 A) ≥ 1, and Then based on expressions (4.5) and (4.6) from condition (4.7) we obtain that for all N ≥ N 0 the following inequalities hold. This means that, owing to the generalized Leray-Schauder type fixed point theorem [18,19,20,21,22,23], the sequence of equations (4.3) has solutions for all N ≥ N 0 , that is all solution setsM N ⊂X N , N ≥ N 0 , are nonempty, and the projectionalgebraic method itself is realizable. Take now ε > 0 and consider the neighborhood It is evident that the closed set D(F)\U ε (M) does not contain solutions to equation (4.1), and for some α ε > 0 the inequality holds. Choose now, based on (4.4), an integer N ε ∈ Z + in such a way that for all N ≥ N ε sup w∈D(F ) Then for all w ∈ D(F)\U ε (M) the following inequality holds, that is for N ≥ N ε there exists the imbeddingM N ⊂ U ε (M). Since ε > 0 is chosen enough small, the conditioñ M N ⊂ U ε (M) for all N ≥ N ε is equivalent to that of convergence for (4.3), proving the theorem.   Proof. It is clear that we need only to verify (4.4). Having assumed contrary, one can find such a subsequence of indices N k ∈ Z + for k ∈ Z + , as well as elements w k ∈ D(F), k ∈ Z + , for which there exists ε > 0, that (4.14) Since for all k ∈ Z + elements f (w k ) ∈ Range(A), owing to the A-compactness of the mapping f : D(F) → Y there exists the limit lim k→∞ f (w k ) = w ∈ Y. Making now use of the existence of limits (4.13), we obtain: contradicting the initial inequality (4.14), thereby proving the theorem.
If the mapping f : D(F) ⊂ X → Y is constant, the operator A : D(A) ⊂ X → Y is densely defined and Range(A) = Y, one can prove additional convergence properties of the projection method of discrete approximations for equation (4.1), to which we proceed below.

Computational scheme convergence analysis: a special degenerate case
Consider the operator problem (4.1), when the mapping F : X → Y is constant, that is F(w) := y ∈ Y for all w ∈ X . Assume that two families of finite-dimensional functional subspacesX N ⊂ X andỸ N ⊂ Y for N ∈ Z + , are chosen such that Assume that a region Ω ⊂ R q is bounded and has a sufficiently smooth boundary ∂Ω, the space X := L p (Ω; R), the dense domain D(A) = W  For further analysis we will need the following convergence proposition for our approximation process.
for each elementṽ N ∈ AX N , N ∈ Z + ; iii) the upper limit Proof. It is easy to see that the equation P (y) N Aw = g, N ∈ Z + , has solutionsw N ∈X N , for which Aw N − g Y → 0 as N → ∞. Then owing to the inequality one can infer that lim N →∞ ρ(g, AX N ) = 0, that is the family of subsets AX N ∈ Y : N ∈ Z + is limiting-dense in Y. Define now N ∈ Z + and consider P  Let noww N ∈X N be the corresponding approximated solution of the equation P N Aw = P N g. Then the following equality Aw N =P (y),−1 N P N g holds, from which and the condition (5.11) one obtains that for any g ∈ Y. But this means that lim N →∞P (y),−1 N P N g = g for every given element g ∈ Y. Making use of the classical Banach-Steinhaus theorem [12,4,24] we obtain that for some bounded value c (y) ∈ R + . Thus for each element P (y) Nw N =w N one finds that 15) where j N :Ỹ N → Y is the corresponding densely defined imbedding operator and j N is its norm. Inequality (5.15) means that the norm of the operatorP where quantities τ (y) N > 0 are bounded for all N ∈ Z + . But this means that the condition ii) of our Statement is fulfilled concerning each elementṽ N ∈ AX N , that is P (y) Sufficiency of conditions i) − iii) shall next be proved as follows. Let us solve the equation P N Aw = P N g for N ∈ Z + , whosew N ∈X N , whose solution is unique. Then they can be represented as where, as above, the linear mappingP for all N ∈ Z + . When for any elementw N ∈ AX N we obtain where we took into account thatP (y),−1 N P (y) Nw N =w N for allw N ∈ AX N . But owing to the assumption iii) this means the existence of the limit lim N →∞ Aw N − g Y = 0 for an arbitrary element g ∈ Y, finishing the proof. Remark 5.3. We note here that an analogous alternative of Proposition 5.2 was earlier proved in [14].
As an obvious corollary of the proof of Proposition 5.2 in the case when dimX N = dimỸ N < ∞ for all N ∈ Z + we obtain that condition i) in form (5.6) follows from ii). Moreover, the next statement about the convergence of the solutionsw N ∈X N as N → ∞ to element w ∈ X holds. Proposition 5.4. Let all the conditions of Proposition 5.2 be fulfilled, in particular, the invertible operator A : X → Y is closed and surjective (this means that A −1 < ∞ owing to the classical statement [12,24,4] about the closed everywhere defined operator). Then the sequence of solutionsw N ∈X N to the equation P Proof. Assume that w N ∈ X N is a solution to the equation P (y) N Aw N = P (y) N g for all N ∈ Z + . Then one can estimate the difference (w −w N ) ∈ X subject to the norm in the Banach space X : Based now on inequality (5.18) we obtain that lim N →∞ Aw N −g Y = 0. As the inverse operator A −1 is closed and everywhere defined and bounded, the right hand side of inequality (5.19) tends to zero as N → ∞. Thereby we state that lim N →∞ w N − w X = 0, completing the proof.

Conclusion
We provided a precise mathematical framework for the class of control problems with delays under investigation in a Banach functional space setting, followed by careful definitions of the key properties to be analyzed such as solvability and complete controllability. The control problem was recast in a reduced form that is especially amenable to the rather innovative analytical approach that we employed. The solvability and completeness of the (reduced) nonlinearly perturbed linear control problem with delay parameters was studied in detail using a Borsuk-Ulam type fixed point theorem to analyze the general topological structure of a suitably reduced control problem solution, focused on estimating the dimension of the corresponding solution set and proving its completeness. Moreover, we investigated the related classical analytical solvability under some special, mildly restrictive, conditions imposed on the linear control and nonlinear functional perturbation. For the application of our approach, we described a new computational projection-based discretization scheme for obtaining accurate approximate solutions of the control problem along with useful error estimates. The scheme effectively reduced the infinite-dimensional problem to a sequence of solvable finite-dimensional matrix valued tasks. In addition, we included an application of the scheme to a special degenerate case wherein the Banach-Steinhaus theorem was brought to bear in the estimation process.