A causal input᎐output system operating for all time from the indefinite past to the indefinite future may be described by a function space for inputs, a function space for outputs, and a causal operator mapping the input space into the output space. The state of such a system at any instant is defined here as an operator from the space of possible future inputs to that of future outputs. This operator is called the natural state. The output space is taken to be a time-shift-invariant normed linear function space, and the input space is either also such a space or a time-shift-invariant subset thereof. There is flexibility allowed in the choice of these spaces. Both the input᎐output operator and the operator giving the natural state are themselves taken to be elements of normed linear spaces with one of a particular family of norms called N-power norms. The general development applies to nonlinear and time-varying systems. Continuity and boundedness of the natural Ž . state as an operator and properties of the natural state and its trajectory as related to the input᎐output description of the system are investigated. Two examples are presented. ᮊ

An alternative solution to the ^-guidance dynamic equations is presented in this paper. The /^-guidance is a midcourse guidance algorithm that maximizes intercept velocity. The nonlinear ^-guidance dynamic equations are linearized. The linearization is via a nonlinear coordinate transformation and nonlinear feedback. The cost function is approximated by a quadratic cost. The linear quadratic regulator (LQR) method is then used to solve the optimal control problem. A closed-form solution to the LQR problem is derived for both the free and fixed final angle of approach cases. A closed-form solution has not been derived for the original solution. A simulation is conducted to compare the trajectories and costs of the linearized solution versus the original solution. The results show similar performance between the two methods. Nomenclature R = range to predicted intercept point Y = velocity vector angle Yf = terminal velocity vector angle constraint 8 = heading error angle K = curvature of trajectory CD = coefficient depending on aerodynamic parameters, a constant

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A b s t met 2. Dynamical EquationsThe 6-guidance dynamical equations, which are nonlinear and vary with respect to the independent parameter, range, are linearized and then the LQR method is used to solve the optimal control problem.The linearization is via a nonlinear coordinate transformation and nonlinear feedback. The cost function is approximated by a quadratic cost.