Controllability and speed of the process described by a parabolic equation with bounded control

To the solution of a boundary value problem with a parameter for an ordinary differential equations

“…This is the principal difference of the proposed method in comparison with known methods. This work is a continuation of the studies presented in [6][7][8][9][10][11][12].…”

“…, ( , , )) is continuous w.r.t. variables ( , , ) ∈ R × R × and satisfies Lipschitz condition in variable , i.e., | ( , , ) − ( , , )| ( )| − |, ∀( , , ), ( , , ) ∈ R × R × , (7) and the condition | ( , , )| 0 (| | + | | 2 ) + 1 ( ), ∀( , , ), (8) where ( ) 0, ( ) ∈ 1 ( , R 1 ), 0 = > 0, 1 ( ) 0, 1 ( ) ∈ 1 ( , R 1 ). We note that under conditions (7), (8), for fixed 0 = ( 0 ) ∈ R , ∈ R differential equation (1) has the unique solution for ∈ .…”

“…The proof of Theorems 1, 2 was given in works [6,7]. Application of Theorems 1, 2 to control problems was presented in [8][9][10], while the boundary value problems for ordinary differential equations were discussed in [11,12].…”

Constructive method for solving a boundary value problem for ordinary differential equations

“…Applications of Theorems 1 and 2 to the solution of controllability and optimal control problems are described in [4][5][6][7].…”

“…As follows from the statements of the problems, one should prove the existence of a pair (x 0 , x 1 ) ∈ S such that the solution of system (1) issuing from the point x 0 at time t 0 passes through the point x 1 at time t 1 ; the state constraint (3) should be satisfied along the solution of system (1) at each time, and the integrals (6) should satisfy conditions (4) and (5). In particular, the set S is defined by the relation S = {(x 0 , x 1 ) ∈ R 2n | H j (x 0 , x 1 ) ≤ 0, j = 1, .…”