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This article develops a model-free adaptive optimal control policy for discrete-time Markov jump systems. First, a two-player zero-sum game is formulated to obtain an optimal control policy that minimizes a cost function against the worst-case disturbance. Second, an action and mode-dependent value function is set up for zero-sum game to search such a policy with convergence guarantee rather than solving an optimization problem satisfying coupled algebraic Riccati equations. To be specific, motivated by the Bellman optimal principle, we develop an online value iterations algorithm to solve the zero-sum game, which is learning while controlling without any initial stabilizing policy. By this algorithm, we can achieve disturbance attenuation for Markov jump systems without knowledge of the system matrices. The adaptivity to slowly changing uncertainties can also be achieved due to the model-free feature and policy convergence. Finally, the effectiveness and practical potential of the algorithm are demonstrated by considering two numerical examples and a solar boiler system.

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Wave diffraction is a typical problem that is encountered in breakwater construction because a knowledge of diffraction behavior has important application in the design and location of breakwaters. With the rapid development of harbor, more and more unconventional layout forms of breakwaters are often encountered in actual engineering. Therefore, this paper introduces a mathematical model of regular waves diffracted by symmetrically arranged rigid and thin breakwaters with a gap, using the conformal transformation method and Green's function method; an appropriate numerical method is described to obtain the approximate analytical solutions. Wave diffraction coefficients were calculated for cases involving different incident angles of the waves, different gap widths, and the included angles between two breakwaters, and the diffraction coefficient diagrams were plotted. The results of the present mathematical model were compared with the those of previous analytical solutions. It was found that the variation trends of the present results are consistent with those of previous studies, which proves the correctness of the model. The influence of the included angle between two breakwaters on the distribution of diffraction coefficients was also analyzed. The mathematical model is unity of the Penny and Price (1952)'s and Kirby et al.(1994)'s models; it effectively overcomes the insufficient accuracy of the superposition method for cases involving oblique incidence or gap widths of less than one wavelength. Furthermore, Lamb (1932)'s solution is reproduced when the ratio of the gap width to the wavelength is very small.

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