. Control theory with applications to naval hydrodynamics. Control theory; Calculus of variations; Dynamic programming. Hence 0 ! r >i J {[F - q' - a'f ] 6x + [F - £'f ] 6u} da + q' (T)Sx(T) + GxAxT + G^T + y' (MxAxT + M^T) ⦠/ T+AT F(a, x, u) + q/x - q_'f(a, x, u) - F(T, xu) â¢T -T' - £'x(T) + q*f(T, xT, uT) da + [F(T, xT, uT) + £'zT + q'6x(T) - q'f(T, xT u,,) ] AT () The integral from T to T+AT is a second order contribution which goes to zero faster than the other terms as AT ->â 0. In order to determine Ax, consider the solutions of the differ- ential Equation (), which ha
. Control theory with applications to naval hydrodynamics. Control theory; Calculus of variations; Dynamic programming. Hence 0 ! r >i J {[F - q' - a'f ] 6x + [F - £'f ] 6u} da + q' (T)Sx(T) + GxAxT + G^T + y' (MxAxT + M^T) ⦠/ T+AT F(a, x, u) + q/x - q_'f(a, x, u) - F(T, xu) â¢T -T' - £'x(T) + q*f(T, xT, uT) da + [F(T, xT, uT) + £'zT + q'6x(T) - q'f(T, xT u,,) ] AT () The integral from T to T+AT is a second order contribution which goes to zero faster than the other terms as AT ->â 0. In order to determine Ax, consider the solutions of the differ- ential Equation (), which have the initial value These solutions satisfy the integral equation Then :(t) = Xq + AxT = (x(T) - z(T) r J !(a> E> u) dCT 0 -AT ) + J f(a, x, H ) da = 6x(T) + f AT + 0(e ) where the geometry of the proof is illustrated in Figure T T+AT Figure 1 â Geometry of the Proof 17. Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original Timman, R. (Reinier), 1917-1975; David W. Taylor Naval Ship Research and Development Center. Bethesda, Md. : David W. Taylor Naval Ship Research and Development Center
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