Optimal control for $n× n$ hyperbolic systems involving operators of infinite order

Year, pages: 2002, 409 - 424

In this paper, the following $n× n$ mixed Dirichlet problem of hyperbolic type

$$ ((∂²) / (∂ t²))y_i(\overline{u}) +∑_|α|=0^∞(-1)^|α|a_α D^{2 α}y_i(\overline{u}) +∑_j=1ⁿa_ijy_j(\overline{u}) =f_i+u_i    in  Q , $$

$$ D^ωy_i(\overline{u})=0    on Σ   for |ω|=0,1,2,… , |ω|≤α-1 , i=1,2,…,n , \tag{D} $$

$$ y_i(x,0;\overline{u})=y_i,0(x) ,   ((∂) / (∂ t))y_i(x,0;\overline{u})=y_i,1(x)    in   \Bbb R^N , $$

where $f_i$ are given functions, $\overline{u}=(u_i)_i=1ⁿ$ and $a_ij$ are coefficients matrix such that

$$ a_ij= \cases 1&if i ≥ j , -1&if i<j , \endcases $$

$$ D^α =((∂^|α|) / ((∂ x₁)^α₁…(∂ x_N)^α_N)) ,    α=(α₁,…,α_N) , $$

is a multi-index for differentiation, $|α|=∑_i=1^Nα_i$, will be discussed, which involve infinite order elliptic operator $A$ having the form

$$ Ay_i(\overline{u})=∑_|α|=0^∞(-1)^|α|a_α D^{2 α}y_i(\overline{u}) . $$

The optimality conditions for this system are given. The problem with Neumann conditions also will be formulated.

El-Saify, H., Bahaa, G. 2002. Optimal control for $n× n$ hyperbolic systems involving operators of infinite order. In Mathematica Slovaca, vol. 52, no.4, pp. 409-424. 0139-9918.

El-Saify, H., Bahaa, G. (2002). Optimal control for $n× n$ hyperbolic systems involving operators of infinite order. Mathematica Slovaca, 52(4), 409-424. 0139-9918.