Though great progress has been made in the Aspect-Based Sentiment Analysis(ABSA) task through research, most of the previous work focuses on English-based ABSA problems, and there are few efforts on other languages mainly due to the lack of training data. In this paper, we propose an approach for performing a Cross-Lingual Aspect Sentiment Classification (CLASC) task which leverages the rich resources in one language (source language) for aspect sentiment classification in a under-resourced language (target language). Specifically, we first build a bilingual lexicon for domain-specific training data to translate the aspect category annotated in the source-language corpus and then translate sentences from the source language to the target language via Machine Translation (MT) tools. However, most MT systems are general-purpose, it non-avoidably introduces translation ambiguities which would degrade the performance of CLASC. In this context, we propose a novel approach called Reinforced Transformer with Cross-Lingual Distillation (RTCLD) combined with target-sensitive adversarial learning to minimize the undesirable effects of translation ambiguities in sentence translation. We conduct experiments on different language combinations, treating English as the source language and Chinese, Russian, and Spanish as target languages. The experimental results show that our proposed approach outperforms the state-of-the-art methods on different target languages.