Absfraci-Recently, attention has been focused on the class of bilinear systems, both for its applicative interest and intrinsic simplicity.In fact, it appears that many important processes, not only in engineering, but also in biology, socio-economics, and ecology, may be modeled by bilinear systems. Moreover, since their nonlinearity is due to products between input and state variables, this class frequently may be studied by techniques similar to those employed for linear systems. This work is intended to motivate the interest of bilinear systemsand to present the current state of research in its various aspects.After an introductory section, in which theoretical and applicative aspects of bilinear systems are enlightened, four other sections follow, respectively, devoted to structural properties, mathematical models, identification and optimization. In a final section, some concluding remarks are made on still open problems and possible trends for future research.

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We address the problem of finding the optimal radiotherapy fractionation scheme, representing the response to radiation of tumour and normal tissues by the LQ model including exponential repopulation and sublethal damage due to incomplete repair. We formulate the nonlinear programming problem of maximizing the overall tumour damage, while keeping the damages to the late and early responding normal tissues within a given admissible level. The optimum is searched over a single week of treatment and its possible structures are identified. In the two simpler but important cases of absence of the incomplete repair term or of prevalent late constraint, we prove the uniqueness of the optimal solution and we characterize it in terms of model parameters. The optimal solution is found to be not necessarily uniform over the week. The theoretical results are confirmed by numerical tests and comparisons with literature fractionation schemes are presented.

A respiratory chamber is used for monitoring O(2) consumption (Vo(2)), CO(2) production (Vco(2)), and respiratory quotient (RQ) in humans, enabling long term (24-h) observation under free-living conditions. Computation of Vo(2) and Vco(2) is currently done by inversion of a mass balance equation, with no consideration of measurement errors and other uncertainties. To improve the accuracy of the results, a new mathematical model is suggested in the present study explicitly accounting for the presence of such uncertainties and error sources and enabling the use of optimal filtering methods. Experiments have been realized, injecting known gas quantities and estimating them using the proposed mathematical model and the Kalman-Bucy (KB) estimation method. The estimates obtained reproduce the known production rates much better than standard methods; in particular, the mean error when fitting the known production rates is 15.6 +/- 0.9 vs. 186 +/- 36 ml/min obtained using a conventional method. Experiments with 11 humans were carried out as well, where Vo(2) and Vco(2) were estimated. The variance of the estimation errors, produced by the KB method, appears relatively small and rapidly convergent. Spectral analysis is performed to assess the residual noise content in the estimates, revealing large improvement: 2.9 +/- 0.8 vs. 3,440 +/- 824 (ml/min)(2) and 1.8 +/- 0.5 vs. 2,057 +/- 532 (ml/min)(2), respectively, for Vo(2) and Vco(2) estimates. Consequently, the accuracy of the computed RQ is also highly improved (0.3 x 10(-4) vs. 800 x 10(-4)). The presented study demonstrates the validity of the proposed model and the improvement in the results when using a KB estimation method to resolve it.