A particular attention is given to the popular array configurations which are the concentric uniform circular-based arrays, cross and square-based centro-symmetric arrays which satisfy these conditions.
By a solution of the initial boundary value problem of the micropolar thermoelasticity in the cylinder (Btimes [0,T)) we mean an ordered array ({ u_{i}, varphi_{i},theta }) which satisfies system (16) for all ((x,t in Btimes [0,T)), the initial conditions (9) and the boundary conditions (10). Throughout this paper it is assumed that a solution ({ u_{i}, varphi _{i},theta }) exists.
Given these assumptions there exists a first-best investment level a FB,b FB that maximizes the overall surplus F a,b −a−b, which satisfies: begin{array}rcl@ frac{partial F}{partial a^{text{FB}}}=frac{partial F}{partial b^{text{FB}}}=1 end{array} (3).
The ellipsoid (mathcal {E}(P)) is an invariant set of the system (37) if and only if there exists a scalar α∈[0,1−ρ 2(A)] which satisfies left[ begin{array}{cc} A^{T}PA- 1-alpha)P &A^{T}PA- 1-alpha& B^{T}PA- 1-alphaend{array} right] Preceq {0} (38).
if and only if there exists a positive definite matrix P which satisfies left[ begin{array}{cc} A^{T}PA-P+C^{T}C &A^{T}PB+C^{T}D B^{T}PA+D^{T} C & B^{T}PB+D^{T}D-mu_{infty}I end{array} right] prec{0}.
(2.6) By deriving the optimality conditions of (2.1), we can easily find that solving (2.1) is equivalent to finding a pair of ((x^,y^;lambda^)) which satisfies left { begin{array}{l} x^inmathcal{X}, quad (x-x^)^{T}(theta(x^ -A^{T}lambda^)geq{0},quad forall{x^ -A^{hcal{X}, y^inmaT}lambda^ geq{0}y-y^)^{T}(gamma(y^)-B^{T}lambda^)geq{0},quad forall{x}inmathcal{X}, Ax^+By^inmathcal{arraY} right.
Now, anything which satisfies PPP satisfies CPP.
which satisfies condition (5.1).
which satisfies the property.
