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Optimal control of linear retarded systems in Banach spaces. (English) Zbl 0603.49005
Consider the linear control problem with retarded arguments $dx(t)/dt=A_ 0x(t)+\int^{0}_{-h}d\eta (s)x(s+t)+f(t)+B(t)u(t)\quad a.e.\quad 0\leq t\leq T,$ $x(0)=g^ 0,\quad x(s)=g^ 1(s)\quad a.e.\quad s\in [-h,0).$ Here the state x(t) lies in a Banach space X, the control u(t) in a Banach space Y, $$A_ 0$$ generates a strongly continuous semigroup on X, and the Stieltjes measure $$\eta$$ is given by $\int^{0}_{-h}d\eta (s)x(s+t)=\sum^{m}_{r=1}A_ rx(t-h_ r)+\int^{0}_{-h}D(s)x(t+s)ds,$ where $$D(\cdot)\in L_{p'}([- h,0];{\mathcal L}(X))$$. The initial condition $$g=(g^ 0,g^ 1)$$ is given with $$g^ 0\in X$$, $$g^ 1\in L_ p([-h,0];X)$$. The author first studies the general linear system (without control), establishes existence and uniqueness of a mild solution, and derives a variation of constants formula for the mild solution in terms of the fundamental solution G(t), which is an operator in $${\mathcal L}(X)$$. Conditions are found under which a mild solution is a strong solution. The author now studies the optimal control problem with cost $J=\phi_ 0(x(T))+\int^{T}_{0}[f_ 0(x(t),t)+k_ 0(u(t),t)]dt$ where $$\phi_ 0:X\to R$$, $$f_ 0:X\times I\to R$$, $$k_ 0:Y\times I\to R$$, $$I=[0,T]$$, and where the admissible control set is closed and convex in $$L_ p(I;Y)$$. The adjoint system and fundamental solution are defined, and existence of an optimal control is proved under convexity hypotheses on $$\phi_ 0,f_ 0,k_ 0$$. Necessary conditions for optimality are obtained, characterized by the solution of the adjoint retarded system. A pointwise maximum principle is established for time-varying admissible control set and examples are given. The bang-bang principle is studied for the time optimal problem.
Reviewer: K.Cooke

MSC:
 49J27 Existence theories for problems in abstract spaces 49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 49K27 Optimality conditions for problems in abstract spaces 46B99 Normed linear spaces and Banach spaces; Banach lattices 93C05 Linear systems in control theory 93C25 Control/observation systems in abstract spaces 49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
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References:
 [1] Ahmed, N.U; Teo, K.L, Optimal control of distributed parameter systems, (1981), North-Holland New York · Zbl 0472.49001 [2] Artola, M, Equations paraboliques à retardement, C. R. acad. sci. Paris, 264, 668-671, (1967) · Zbl 0145.36001 [3] Balakrishnan, A.V, Optimal control problems in Banach spaces, SIAM J. control, 3, 152-180, (1965) · Zbl 0178.44601 [4] Balakrishnan, A.V, Applied functional analysis, (1981), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0459.46014 [5] Banks, H.T, Necessary conditions for control problems with variable time-lags, SIAM J. control, 6, 9-47, (1969) · Zbl 0159.13002 [6] Banks, H.T; Jacobs, M.Q, An attainable sets approach to optimal control of functional differential equations with function space boundary conditions, J. differential equations, 13, 127-149, (1973) · Zbl 0256.49044 [7] Banks, H.T; Kent, G.T, Control of functional differential equations of retarded and neutral type with target sets in function space, SIAM J. control, 10, 567-593, (1972) · Zbl 0266.49014 [8] Banks, H.T; Manitius, A, Application of abstract variational theory to hereditary systems—A survey, IEEE trans. automat. control, AC-19, 524-533, (1974) · Zbl 0288.49004 [9] Barbu, V, Nonlinear semi-groups and differential equations in Banach spaces, (1976), Noordhoff Leyden, the Netherlands [10] Bien, Z; Chyung, D.H, Optimal control of delay systems with a final function condition, Internat. J. control, 32, 539-560, (1980) · Zbl 0465.49010 [11] Chyung, D.H; Lee, E.B, Linear optimal systems with time delays, SIAM J. control, 4, 548-575, (1966) · Zbl 0148.33804 [12] Colonius, F, The maximum principle for relaxed hereditary differential systems with function space end condition, SIAM J. control optim., 20, 695-712, (1982) · Zbl 0525.49017 [13] Colonius, F; Hinrichsen, D, Optimal control of functional differential systems, SIAM J. control optim., 16, 861-879, (1978) · Zbl 0438.49015 [14] Curtain, R.F; Pritchard, A.J, Infinite dimensional linear systems theory, () · Zbl 0352.49003 [15] Delfour, M.C, The linear quadratic optimal control problem for hereditary differential systems: theory and numerical solution, Appl. math. optim., 3, 101-162, (1977) · Zbl 0404.49010 [16] Delfour, M.C; Mitter, S.K, Controllability, observability and optimal feedback control of affine hereditary differential systems, SIAM J. control, 10, 298-328, (1972) · Zbl 0242.93011 [17] Dunford, N; Schwartz, J.T, Linear operators, part I, (1966), Interscience New York · Zbl 0146.12601 [18] Friedman, A, Optimal control in Banach spaces, J. math. anal. appl., 19, 35-55, (1967) · Zbl 0155.15902 [19] Gibson, J.S, Linear-quadratic optimal control of hereditary differential systems: infinite dimensional ricati equations and numerical approximations, SIAM J. control optim., 21, 95-139, (1983) · Zbl 0557.49017 [20] Hale, J.K, Theory of functional differential equations, (1977), Springer-Verlag New York/Heidelberg/Berlin · Zbl 0425.34048 [21] Hille, E; Phillips, R.S, Functional analysis and semi-group, () [22] Jacobs, M.Q; Kao, T.J, An optimum setting problem for time lag systems, J. math. anal. appl., 40, 687-707, (1972) · Zbl 0211.46403 [23] Kato, T, Accretive operators and non-linear evolution equations in Banach spaces, (), 138-161 [24] Kato, T, Perturbation theory for linear operators, (1976), Springer-Verlag Berlin/Heidelberg/New York [25] Komatsu, H, Semi-groups of operators in locally convex spaces, J. math. soc. Japan, 16, 230-262, (1964) · Zbl 0154.16103 [26] Lions, J.L, Optimal control of systems governed by partial differential equations, (1971), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0203.09001 [27] Nababan, S; Teo, K.L, On the existence of optimal controls of the first boundary value problems for parabolic delay-differential equations in divergence form, J. math. soc. Japan, 32, 343-362, (1980) [28] Nababan, S; Teo, K.L, Necessary conditions for optimality of Cauchy problems for parabolic partial delay-differential equations, J. optim. theory appl., 34, 117-155, (1981) · Zbl 0431.49024 [29] Nakagiri, S, On the fundamental solution of delay-differential equations in Banach spaces, J. differential equations, 41, 349-368, (1981) · Zbl 0441.35068 [30] \scS. Nakagiri, Pointwise completeness and degeneracy of functional differential equations in Banach spaces, I, II, to appear. · Zbl 0648.34080 [31] Oguztöreli, M.N, Time-lag control systems, (1966), Academic Press New York · Zbl 0143.12101 [32] Phillips, R.S, The adjoint semi-group, Pacific J. math., 5, 269-283, (1955) · Zbl 0064.11202 [33] Tanabe, H, Equations of evolution, (1979), Pitman New York [34] Teo, K.L, Optimal control of systems governed by time delayed second order linear parabolic partial differential equations with a first boundary condition, J. optim. theory appl., 29, 437-481, (1979) · Zbl 0387.49012 [35] Wang, P.K.C, Optimal control of parabolic systems with boundary conditions involving time delays, SIAM J. control, 13, 274-293, (1975) · Zbl 0301.49009 [36] Warga, J, Optimal control of differential and functional equations, (1972), Academic Press New York · Zbl 0253.49001
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