## 5 Conclusions  
In this paper, we apply the direct-comparison based optimization approach to study the rewards optimization of the discrete-time stochastic linear-quadratic control problem with conic constraints on an infinite horizon. We derive the performance difference formula by utilizing the state separation property of the system structure. Based on this, the optimality condition and the stationary optimal feedback control can be obtained. The direct-comparison based approach is applicable to both linear and nonlinear systems. By introducing the the LQ optimization problem, we establish a general framework for studying infinite horizon control problems with total rewards. We verify that the proposed optimal approach can solve the LQ problems. Then we illustrate our results by two simulation examples. The results can easily be extended to the cases of non-Gaussian noises and average rewards. Most significantly, our methodology can deal with a very general class of linear constraints on state and control variables, which includes the cone constraints, positivity and negativity constraints, and the state-dependent upper and lower bound constraints as a special case. In addition to the problem with the infinite control horizon, our results still fit problems with a finite horizon. In addition, without identifying all the system structure parameters, this approach can also be implemented on-line, and learning based algorithms can be developed. Finally, this work focuses on the discrete-time stochastic LQ control problem. Our next step is to investigate continuous cases. As the constrained LQ problem has a wide range of applications, we hope to apply our approach in more areas, such as dynamic portfolio management, security optimization of cyber-physical systems, and financial derivative pricing, in our future research. Conceptualization, R.X. and X.Y. ; methodology, R.X. ; validation, R.X., X.Y. and W.W.; formal analysis, X.Y. ; data curation, X.Y. ; writing-original draft preparation, R.X. and X.Y. ; writing-review and editing, R.X., X.Y. and W.W.; supervision, W.W. All authors have read and agreed to the published version of the manuscript. This work was supported by the National Natural Science Foundation of China under Grant 61573244 \"Stochastic control optimization of uncertain systems based on offset with multiplicative noises and its applications in the financial optimization\" and 61521063 \"Control theory and techniques: design, control and optimization of network systems\". The authors declare no conflict of interest. The following abbreviations are used in this manuscript:  
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