ba756f9f-4f30-46ca-9e54-3e8389a58dd720250128083812986wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON SYSTEMS AND CONTROL2224-28561991-876310.37394/23203http://wseas.org/wseas/cms.action?id=40731720251720252010.37394/23203.2025.20https://wseas.com/journals/sac/2025.phpMarioLefebvreDepartment of Mathematics and Industrial Engineering, Polytechnique Montréal, 2500, chemin de Polytechnique, Montréal (Québec) H3T 1J4, CANADALet {Y (t), t ≥ 0} be a controlled geometric Brownian motion and X(t) be the integral of Y (t). The problem of minimizing the expected time that the ratio X(t)/Y (t) will spend between two constants is considered. The optimal control is obtained explicitly in terms of special functions. A risk-sensitive version of the cost criterion is also proposed.128202512820259122https://wseas.com/journals/sac/2025/a045103-002(2025).pdf10.37394/23203.2025.20.2https://wseas.com/journals/sac/2025/a045103-002(2025).pdf10.1109/9.83548R. Rishel, Controlled wear process: modeling optimal control, IEEE Transactions on Automatic Control, Vol. 36, No. 9, 1991, pp. 1100–1102. https://doi.org/10.1109/9.83548 P. Whittle, Optimization over Time, Vol. I, Wiley, Chichester, 1982. 10.3390/fractalfract7020152M. Lefebvre, First-passage times and optimal control of integrated jump-diffusion processes, Fractal and Fractional, Vol. 7, No. 2, 2023, Article 152. https://doi.org/10.3390/fractalfract7020152 10.1007/s11009-020-09800-2M. Kounta and N.J. Dawson, Linear quadratic Gaussian homing for Markov processes with regime switching and applications to controlled population growth/decay, Methodology and Computing in Applied Probability, Vol. 23, 2021, pp. 1155–1172. https://doi.org/10.1007/s11009-020-09800-2 10.1109/tac.2022.3157077C. Makasu, Homing problems with control in the diffusion coefficient, IEEE Transactions on Automatic Control, Vol. 67, No. 7, 2022, pp. 3770–3772. https://doi.org/10.1109/ TAC.2022.3157077 10.1016/j.automatica.2024.111575C. Makasu, Bounds for a risk-sensitive homing problem, Automatica, Vol. 163, 2024, 111575. https://doi.org/10.1016 /j.automatica.2024.111575 R. Bellman, Dynamic programming, Princeton University Press, Princeton, NJ, USA, 1957. 10.2307/1266136M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1965. P. Whittle, Risk-Sensitive Optimal Control, Wiley, Chichester, 1990. 10.2307/3213947J. Kuhn, The risk-sensitive homing problem, Journal of Applied Probability, Vol. 22, No. 4, 1985, pp. 796–803. https://doi.org/10.2307/3213947.