A priori estimates for the optimal control of an oscillating water column. (English) Zbl 0761.76014
Summary: The results of this paper are estimates for periodic solutions of the equations of one-dimensional motion of a floating body. The equations are the state equations in a control problem, and the goal is to obtain estimates independent of the control. We solve an abstract problem which contains the model problem described in [R. E. Hoskin, B. M. Count, N. K. Nichols and D. A. C. Nicol, in Hydronamics of ocean wave- energy utilization, 257-268 (1985)]. Both initial value and periodic problems are described, where the form of the so-called “added damping” is the same in both problems. It is demonstrated here that the “added dampings” described for the two problems are physically incompatible. A review of the derivation of the damping from physical principles indicates that the model described above must be modified for the periodic problem, and a modification is suggested. The correct formulation greatly simplifies the mathematical problem and gives a natural interpretation to any data collected experimentally in order to measure the “added damping”.
MSC:
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 76M30 | Variational methods applied to problems in fluid mechanics |
| 49N70 | Differential games and control |
| 49N75 | Pursuit and evasion games |
References:
| [1] | Cummins, W. E., Schiffstechnik, 9, 101-109 (1962) |
| [2] | Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1983), Springer: Springer Berlin · Zbl 0691.35001 |
| [3] | Hoskin, R. E.; Count, B. M.; Nichols, N. K.; Nicol, D. A.C., (Hydronamics of Ocean Wave-Energy Utilization (1985), Springer: Springer Berlin), 257-268 |
| [4] | MacCamy, R. C.; Wong, J. S.W., Trans. Am. Math. Soc., 164, 1-37 (1972) · Zbl 0274.45012 |
| [5] | Stein, E. M.; Weiss, G., Introduction to Fourier Analysis on Euclidian Spaces (1971), Princeton University Press · Zbl 0232.42007 |
| [6] | Wehausen, J. V., J. Engng Math., 1, 1, 1-17 (1967) · Zbl 0155.56601 |
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